Deep ReLU neural networks overcome the curse of dimensionality when
  approximating semilinear partial integro-differential equations
In this paper we consider PIDEs with gradient-independent Lipschitz
continuous nonlinearities and prove that deep neural networks with ReLU
activation function can approximate solutions of such semilinear PIDEs without
curse of dimensionality in the sense that the required number of parameters in
the deep neural networks increases at most polynomially in both the dimension $
d $ of the corresponding PIDE and the reciprocal of the prescribed accuracy
$\epsilon $.
Ariel Neufeld, Tuan Anh Nguyen, Sizhou Wu
# Deep ReLU neural networks overcome the curse of dimensionality

###### Abstract.

In this paper we consider PIDEs with gradient-independent Lipschitz continuous nonlinearities and prove that deep neural networks with ReLU activation function can approximate solutions of such semilinear PIDEs without curse of dimensionality in the sense that the required number of parameters in the deep neural networks increases at most polynomially in both the dimension \(d\) of the corresponding PIDE and the reciprocal of the prescribed accuracy \(\epsilon\).

Key words and phrases:Curse of dimensionality, high-dimensional PDEs, high-dimensional partial integro-differential equations, deep neural networks, multilevel Picard approximations, stochastic fixed point equations, stochastic differential equations with jumps 2020 Mathematics Subject Classification: 65C99, 65C05, 65C30, 68T07 Financial support by the Nanyang Assistant Professorship Grant (NAP Grant) _Machine Learning based Algorithms in Finance and Insurance_ is gratefully acknowledged.
and let \(\mathcal{D}\colon\mathbf{N}\to\mathbf{D}\), \(\mathcal{P}\colon\mathbf{N}\to\mathbb{N}\), \(\mathcal{R}\colon\mathbf{N}\to(\cup_{k,l\in\mathbb{N}}C(\mathbb{R}^{k},\mathbb{R} ^{l}))\) satisfy that for all \(H\in\mathbb{N}\), \(k_{0},k_{1},\ldots,k_{H},k_{H+1}\in\mathbb{N}\), \(\Phi=((W_{1},B_{1}),\ldots,(W_{H+1},B_{H+1}))\in\prod_{n=1}^{H+1}\left(\mathbb{ R}^{k_{n}\times k_{n-1}}\times\mathbb{R}^{k_{n}}\right),\,x_{0}\in\mathbb{R}^{k_{0}},\ldots,x_{H}\in\mathbb{R}^{k_{H}}\) with the property that \(\forall\,n\in\mathbb{N}\cap[1,H]\colon x_{n}=\mathbf{A}_{k_{n}}(W_{n}x_{n-1}+B _{n})\) we have that_

\[\mathcal{P}(\Phi)=\sum_{n=1}^{H+1}k_{n}(k_{n-1}+1),\quad\mathcal{D}(\Phi)=(k_ {0},k_{1},\ldots,k_{H},k_{H+1}), \tag{3}\]

\(\mathcal{R}(\Phi)\in C(\mathbb{R}^{k_{0}},\mathbb{R}^{k_{H+1}}),\) _and_

\[(\mathcal{R}(\Phi))(x_{0})=W_{H+1}x_{H}+B_{H+1}. \tag{4}\]

Let us comment on the mathematical objects in Setting 1.1. For all \(d\in\mathbb{N}\), \(\mathbf{A}_{d}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}\) refers to the componentwise rectified linear unit (ReLU) activation function. By \(\mathbf{N}\) we denote the set of all parameters characterizing artificial feed-forward DNNs, by \(\mathcal{R}\) we denote the operator that maps each parameters characterizing a DNN to its corresponding function, by \(\mathcal{P}\) we denote the function that counts the number of parameters of the corresponding DNN, and by \(\mathcal{D}\) we denote the function that maps the parameters characterizing a DNN to the vector of its layer dimensions.

### Main result

**Theorem 1.2**.: _Consider the notations in Subsection 1.4, assume Setting 1.1, let \(T,c\in(0,\infty)\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(v\in\mathds{R}^{d}\) let \(\beta_{\varepsilon}^{d}\in C(\mathds{R}^{d},\mathds{R}^{d})\), \(\sigma_{\varepsilon}^{d}\in C(\mathds{R}^{d},\mathds{R}^{d\times d})\), \(\Phi_{\beta_{d}^{d}},\Phi_{\sigma_{\varepsilon}^{d},v}\in\mathbf{N}\) satisfy that \(\beta_{\varepsilon}^{d}=\mathcal{R}(\Phi_{\beta_{\varepsilon}^{d}})\), \(\sigma_{\varepsilon}^{d}(\cdot)v=\mathcal{R}(\Phi_{\sigma_{\varepsilon}^{d},v})\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(\gamma_{\varepsilon}^{d}\colon\mathbb{R}^{2d}\to\mathbb{R}^{d}\), \(F_{\varepsilon}^{d}\colon\mathbb{R}^{d}\to\mathbb{R}^{d\times d}\), \(G^{d}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}\) be measurable and satisfy for all \(y,z\in\mathds{R}^{d}\) that \(\gamma_{\varepsilon}^{d}(y,z)=F_{\varepsilon}^{d}(y)G^{d}(z)\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(v\in\mathds{R}^{d}\) let \(\Phi_{F_{\varepsilon}^{d},v}\in\mathbf{N}\) satisfy \(F_{\varepsilon}^{d}(\cdot)v=\mathcal{R}(\Phi_{F_{\varepsilon}^{d},v})\), assume for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1]\), \(v\in\mathds{R}^{d}\) that \(\mathcal{D}(\Phi_{\sigma_{\varepsilon}^{d},v})=\mathcal{D}(\Phi_{\sigma_{ \varepsilon}^{d},0})\) and \(\mathcal{D}(\Phi_{F_{\varepsilon}^{d},v})=\mathcal{D}(\Phi_{F_{\varepsilon}^{d },0})\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(g_{\varepsilon}^{d}\in C(\mathds{R}^{d},\mathds{R})\), \(\Phi_{g_{\varepsilon}^{d}}\in\mathbf{N}\) satisfy that \(\mathcal{R}(\Phi_{g_{\varepsilon}^{d}})=g_{\varepsilon}^{d}\), for every \(d\in\mathbb{N}\) let \(\beta^{d}\in C(\mathds{R}^{d},\mathds{R}^{d})\), \(\sigma^{d}\in C(\mathds{R}^{d},\mathds{R}^{d\times d})\), \(g^{d}\in C(\mathds{R}^{d},\mathds{R})\), let \(f\in C(\mathds{R},\mathds{R})\), for every \(d\in\mathbb{N}\) let \(\nu^{d}\colon\mathcal{B}(\mathbb{R}^{d}\setminus\{0\})\to[0,\infty)\) be a Levy measure, assume that for all \(d\in\mathbb{N}\) there exists \(C_{d}\in(0,\infty)\) such that for all \(x,y,z\in\mathds{R}^{d}\), \(t\in[0,T]\) we have that_

\[\left\|\gamma^{d}(x,z)\right\|\leq C_{d}\left(1\wedge\|z\|^{2}\right),\quad\left\| \gamma^{d}(x,z)-\gamma^{d}(y,z)\right\|^{2}\leq C_{d}\|x-y\|^{2}\left(1\wedge\| z\|^{2}\right), \tag{5}\]

_assume that for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(x,z\in\mathds{R}^{d}\) the Jacobian matrix \((D_{x}\gamma^{d})(x,z)\) exists, assume that for all \(d\in\mathbb{N}\) there exists \(\lambda_{d}\in(0,\infty)\) such that for all \(t\in[0,T]\), \(x,z\in\mathds{R}^{d}\), \(\delta\in[0,1]\) we have that_

\[\lambda_{d}\leq\left|\det(I_{d}+\delta(D_{x}\gamma^{d})(x,z))\right|, \tag{6}\]

_where \(I_{d}\) denotes the \(d\times d\) identity matrix, and assume for all \(d\in\mathbb{N}\), \(x,y\in\mathds{R}^{d}\), \(w_{1},w_{2}\in\mathbb{R}\), \(\varepsilon\in(0,1)\) that_

\[\left\|\beta_{\varepsilon}^{d}(x)-\beta_{\varepsilon}^{d}(y)\right\|^{2}+\left\| \sigma_{\varepsilon}^{d}(x)-\sigma_{\varepsilon}^{d}(y)\right\|_{\mathrm{F}}^{2} +\int_{\mathds{R}^{d}\setminus\{0\}}\left\|\gamma_{\varepsilon}^{d}(x,z)-\gamma_ {\varepsilon}^{d}(y,z))\right\|^{2}\nu^{d}(dz)\leq c\|x-y\|^{2}, \tag{7}\]

\[|f(w_{1})-f(w_{2})|^{2}\leq c|w_{1}-w_{2}|^{2},\quad\left|g_{ \varepsilon}^{d}(x)-g_{\varepsilon}^{d}(y)\right|^{2}\leq cd^{c}\|x-y\|^{2}, \tag{8}\]

\[\left\|\beta_{\varepsilon}^{d}(0)\right\|^{2}+\left\|\sigma_{ \varepsilon}^{d}(0)\right\|_{\mathrm{F}}^{2}+\int_{\mathds{R}^{d}\setminus\{0 \}}\left\|\gamma_{\varepsilon}^{d}(0,z)\right\|^{2}\nu^{d}(dz)+|f(0)|^{2}+|g_{ \varepsilon}^{d}(0)|^{2}\leq cd^{c}, \tag{9}\]

\[\left\|\beta_{\varepsilon}^{d}(x)-\beta^{d}(x)\right\|^{2}+\left\| \sigma_{\varepsilon}^{d}(x)-\sigma^{d}(x)\right\|_{\mathrm{F}}^{2}+\int_{ \mathds{R}^{d}\setminus\{0\}}\left\|\gamma_{\varepsilon}^{d}(x,z)-\gamma^{d}(x,z )\right\|^{2}\,\nu^{d}(dz)+\left|g_{\varepsilon}^{d}(x)-g^{d}(x)\right|^{2} \tag{10}\] \[\leq ccd^{c}(d^{c}+\|x\|^{2}),\]

_and_

\[\mathcal{P}(\Phi_{\beta_{d}^{d}})+\mathcal{P}(\Phi_{\sigma_{\varepsilon}^{d},0})+ \mathcal{P}(\Phi_{F_{\varepsilon}^{d},0})+\mathcal{P}(\Phi_{g_{d}^{d}})\leq d^{c} \varepsilon^{-c}. \tag{11}\]

_Then_
1. _for every_ \(d\in\mathbb{N}\) _there exists a unique viscosity solution_1__\(u^{d}\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}\) _to the PIDE_ \[\left\{\begin{array}{l}(\frac{\partial}{\partial t}u^{d})(t,x)+\langle\beta^{d }(x),(\nabla_{x}u^{d})(t,x)\rangle\\ +\frac{1}{2}\mathrm{trace}\big{(}\sigma^{d}(t,x)(\sigma^{d}(t,x))^{\top} \mathrm{Hess}_{x}u^{d}(t,x)\big{)}+f(u^{d}(t,x))\\ +\int_{\mathbb{R}^{d}}\Big{(}u^{d}(x+\gamma^{d}(x,z))-u^{d}(t,x)-\Big{\langle} (\nabla_{x}u^{d})(t,x),\gamma^{d}(x,z)\Big{\rangle}\Big{)}\,\nu^{d}(dz)=0\\ \quad\forall\,t\in[0,T),x\in\mathbb{R}^{d},\\ u^{d}(T,x)=g^{d}(x)\quad\forall\,x\in\mathbb{R}^{d}\end{array}\right.\] (12) _satisfying that_ \(\sup_{s\in[0,T],\eta\in\mathbb{R}^{d}}\frac{|u^{d}(s,\eta)|}{1+\|y\|}<\infty\) _and_ F _(_12_)_ _there exist_ \(\eta\in(0,\infty)\) _and_ \((\Psi_{d,\epsilon})_{d\in\mathbb{N},\epsilon\in(0,1)}\subseteq\mathbb{N}\) _such that for all_ \(d\in\mathbb{N}\)_,_ \(\epsilon\in(0,1)\) _we have that_ \(\mathcal{R}(\Psi_{d,\epsilon})\in C(\mathbb{R}^{d},\mathbb{R})\)_,_ \(\mathcal{P}(\Psi_{d,\epsilon})\leq\eta d^{\eta}\epsilon^{-\eta}\)_, and_ F _(_13_)_ \[\left(\int_{[0,1]^{d}}\left|(\mathcal{R}(\Psi_{d,\epsilon}))(x)-u^{d}(t,x) \right|^{2}dx\right)^{\frac{1}{2}}\leq\epsilon.\] (13)

Footnote 1: For the definition of a viscosity solution see, e.g., [31, Definition 2.7].

Let us make some comments on the mathematical objects in Theorem 1.2. The functions \(\mu_{d}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}\), \(d\in\mathbb{N}\), and \(\sigma_{d}\colon\mathbb{R}^{d}\to\mathbb{R}^{d\times d}\), \(d\in\mathbb{N}\), describe the linear part of the family of PIDEs indexed by \(d\in\mathbb{N}\) in (12). The functions \(g_{d}\colon\mathbb{R}^{d}\to\mathbb{R},d\in\mathbb{N},\) describe the initial condition, while the function \(f\colon\mathbb{R}\to\mathbb{R}\) describes the nonlinearity of the PIDEs in (12). Condition (5) and (6) are needed for the existence and uniqueness of the solution to the PIDE (12) (see the proof of Theorem 5.1). Conditions (7)-(8) are globally Lipschitz condition. Condition (9) states that the initial values of the input functions grow polynomially in the dimension \(d\in\mathbb{N}\). Condition (10) ensures that the input functions \(\beta^{d},\sigma^{d},\gamma^{d},g^{d}\) can be approximated by the functions \(\beta^{d}_{\varepsilon},\sigma^{d}_{\varepsilon},\gamma^{d}_{\varepsilon},g^{d}_ {\varepsilon}\). The bound \(d^{\varepsilon}e^{-c}\), which is a polynomial of \(d\) and \(\varepsilon^{-1}\), in condition (11) ensures that the functions \(\beta^{d}_{\varepsilon},\sigma^{d}_{\varepsilon},\gamma^{d}_{\varepsilon},g^{d}_ {\varepsilon}\) can be represented by DNNs without curse of dimensionality. Under these assumptions Theorem 1.2 states that, roughly speaking, if DNNs can approximate the initial condition, the linear part, the nonlinearity, and the jump part of the PIDEs in (12) without the curse of dimensionality, then they can also approximate its solution without the curse of dimensionality. We refer to [1, 7, 8, 19, 24, 30] for similar results obtained for PDEs _without any non-local/ jump term_.

### Outline of the proof and organization of the paper

Theorem 1.2 follows directly from Theorem 5.1, see the proof of Theorem 1.2 which is provided right after the proof of Theorem 5.1. Although the result presented in Theorem 1.2 is purely deterministic, we use probabilistic arguments to prove its statement. More precisely, we employ the theory of full history recursive MLP approximations, which are numerical approximation methods for which it is known that they overcome the curse of dimensionality. We refer to [31] for the convergence analysis of MLP algorithms for semilinear PIDEs and to [3, 4, 12, 15, 21, 22, 23, 25, 26, 27, 28] for corresponding results proving that MLP algorithms overcome the curse of dimensionality for PDEs without any non-local/ jump term.

The main strategy of the proof, roughly speaking, is to demonstrate that these MLP approximations can be represented by DNNs, if the coefficients determining the linear part, the jump term, the initial condition, and the nonlinearity are represented by DNNs (cf. Lemma 4.12). Such ideas have been successfully applied to prove that DNNs overcome the curse of dimensionality in the numerical approximations of _semilinear_ heat equations (see [24]) as well as _semilinear_ Kolmogorov PDEs (see [8]). We also refer to [19, 30] for results proving that DNNs overcome the curse of dimensionality when approximating _linear_ PDEs.

In order to introduce the outline of the proof we first need an MLP setting. For every \(K\in\mathbb{N}\) let \(\lfloor\cdot\rfloor_{K}\colon\mathbb{R}\to\mathbb{R}\) satisfy for all \(t\in\mathbb{R}\) that \(\lfloor t\rfloor_{K}=\max(\{0,\frac{T}{K},\frac{2T}{K},\dots,T\}\cap((-\infty,t )\cup\{0\})\), let \((\Omega,\mathcal{F},\mathbb{P},(\mathbb{F}_{t})_{t\in[0,T]})\) be a probability space satisfying the usual conditions, let \(\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}\), let \(\mathfrak{t}^{\theta}\colon\Omega\to[0,1]\), \(\theta\in\Theta\), be identically independently distributed random variables which satisfy for all \(t\in(0,1)\) that \(\mathbb{P}\,(\mathfrak{t}^{0}\leq t)=t\), for every \(\theta\in\Theta\), \(t\in[0,T]\) let \(\mathfrak{T}^{\theta}_{t}\colon\Omega\to\mathbb{R}\) satisfy for all \(\theta\in\Theta\) that \(\mathfrak{T}^{\theta}_{t}=t+(T-t)\mathfrak{t}^{\theta}\), for every \(d\in\mathbb{N}\) let \(W^{d,\theta}\colon\Omega\times[0,T]\to\mathbb{R}^{d}\), \(\theta\in\Theta\), be identically independently distributed standard \((\mathbb{F}_{t})_{t\in[0,T]}\)-Brownian motions, for every \(d\in\mathbb{N}\) let \(N^{d,\theta}\), \(\theta\in\Theta\), be independent \((\mathbb{F}_{t})_{t\in[0,T]}\)-Poisson random measures on \([0,\infty)\times(\mathbb{R}^{d}\setminus\{0\})\) with intensity \(\nu^{d}\), for every \(d\in\mathbb{N}\), \(\theta\in\Theta\) let

\[\tilde{N}^{d,\theta}(dt,dz)=N^{d,\theta}(dt,dz)-dt\,\nu^{d}(dz), \tag{14}\]
and assume for all \(d\in\mathbb{N}\) that \(\mathcal{F}_{0}\), \((\mathfrak{t}^{\theta})_{\theta\in\Theta}\), \((N^{d,\theta})_{\theta\in\Theta}\) and \((W^{d,\theta})_{\theta\in\Theta}\) are independent. First, the viscosity solution to (12) can be represented by the following stochastic fixed point equation (SFPE) (cf. [31, Proposition 5.16]):

\[u^{d}(t,x)=\mathbb{E}\Big{[}g^{d}(X_{T}^{d,0,t,x})\Big{]}+\int_{t}^{T}\mathbb{ E}\Big{[}f(u^{d}(X_{s}^{d,0,t,x}))\Big{]}\,ds \tag{15}\]

where

\[\begin{split} X_{s}^{d,\theta,t,x}&=x+\int_{t}^{s} \beta^{d}(X_{u-}^{d,\theta,t,x})du\\ &\quad+\int_{t}^{s}\sigma^{d}(X_{u-}^{d,\theta,t,x})dW_{u}^{d, \theta}+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\gamma^{d}(X_{u-}^{d, \theta,t,x},z)\tilde{N}^{d,\theta}(du,dz).\end{split} \tag{16}\]

We approximate the input functions \(\beta^{d},\sigma^{d},\gamma^{d},g^{d},f\) by \(\beta^{d}_{\varepsilon},\sigma^{d}_{\varepsilon},\gamma^{d}_{\varepsilon}, g^{d}_{\varepsilon},f_{\varepsilon}\) where the functions with index \(\varepsilon\) are represented by DNNs. We then get the following SFPE:

\[u^{d,\varepsilon}(t,x)=\mathbb{E}\Big{[}g^{d}_{\varepsilon}(X_{T}^{d,0, \varepsilon,t,x})\Big{]}+\int_{t}^{T}\mathbb{E}\Big{[}f_{\varepsilon}(u^{d, \varepsilon}(X_{s}^{d,0,\varepsilon,t,x}))\Big{]}\,ds \tag{17}\]

where

\[\begin{split} X_{s}^{d,\theta,\varepsilon,t,x}&=x+ \int_{t}^{s}\beta^{d}_{\varepsilon}(X_{u-}^{d,\theta,\varepsilon,t,x})du\\ &\quad+\int_{t}^{s}\sigma^{d}_{\varepsilon}(X_{u-}^{d,\theta, \varepsilon,t,x})dW_{u}^{d,\theta}+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus \{0\}}\gamma^{d}_{\varepsilon}(X_{u-}^{d,\theta,\varepsilon,t,x},z)\tilde{N} ^{d,\theta}(du,dz).\end{split} \tag{18}\]

Next, we approximate the stochastic differential equation (SDE) in (18) by the Euler-Maruyama discretization:

\[\begin{split} X_{s}^{d,\theta,K,\varepsilon,t,x}&=x+ \int_{t}^{s}\beta^{d}_{\varepsilon}(X_{\max\{t,[u]-\rfloor_{K}\}}^{d,\theta,K, \varepsilon,t,x})du+\int_{t}^{s}\sigma^{d}_{\varepsilon}(X_{\max\{t,[u]- \rfloor_{K}\}}^{d,\theta,K,\varepsilon,t,x})dW_{u}^{d,\theta}\\ &\quad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\gamma^{d} _{\varepsilon}(X_{\max\{t,[u]-\rfloor_{K}\}}^{d,\theta,K,\varepsilon,t,x}) \,\tilde{N}^{d,\theta}(du,dz).\end{split} \tag{19}\]

The latter is associated to the following SFPE:

\[u^{d,K,\varepsilon}(t,x)=\mathbb{E}\Big{[}g^{d}_{\varepsilon}(X_{T}^{d,0,K, \varepsilon,t,x})\Big{]}+\int_{t}^{T}\mathbb{E}\Big{[}f_{\varepsilon}(u^{d, K,\varepsilon}(X_{s}^{d,0,K,\varepsilon,t,x}))\Big{]}\,ds. \tag{20}\]

The DNNs that approximate \(u^{d}\), \(d\in\mathbb{N}\), will be constructed from the following MLP approximation:

\[\begin{split} U_{n,m}^{d,\theta,K,\varepsilon}(t,x)& =\frac{1_{\mathbb{N}}(n)}{m^{n}}\sum_{i=1}^{m^{n}}g^{d}_{\varepsilon}(X_{T}^{d,(\theta,0,-i),K,\varepsilon,t,x})\\ &\quad+\sum_{\ell=0}^{n-1}\frac{T-t}{m^{n-\ell}}\sum_{i=1}^{m^{n} }(f\circ U_{\ell,m}^{d,(\theta,\ell,i),K,\varepsilon}-f\circ U_{\ell-1,m}^{d,( \theta,-\ell,i),K,\varepsilon})(\overline{x}_{t}^{(\theta,\ell,i)},X_{ \overline{x}_{t}^{(\theta,\ell,i)}}^{d,(\theta,0,i),K,\varepsilon,t,x})\end{split} \tag{21}\]

(cf. Lemma 4.12). We then decompose the error \(U_{n,m}^{d,\theta,K,\varepsilon}(t,x)-u^{d}(t,x)\) as follows:

\[\begin{split} U_{n,m}^{d,\theta,K,\varepsilon}(t,x)-u^{d}(t,x)\\ &=\underbrace{U_{n,m}^{d,\theta,K,\varepsilon}(t,x)-u^{d,K, \varepsilon}(t,x)}_{=:E_{1}}+\underbrace{u^{d,K,\varepsilon}(t,x)-u^{d, \varepsilon}(t,x)}_{=:E_{2}}+\underbrace{u^{d,\varepsilon}(t,x)-u^{d}(t,x)}_{=: E_{3}}\end{split} \tag{22}\]

where \(E_{1}\) is estimated in Lemma 3.3, \(E_{2}\) is estimated in Lemma 3.2, and \(E_{3}\) is estimated in Lemma 2.1. In the proof of our main result, Theorem 5.1, we combine these results and construct a DNN realization on a probability space that suffices the prescribed approximation accuracy \(\epsilon\in(0,1)\) between \(U_{n,m}^{d,\theta,K,\varepsilon}(t,x)\) and \(u^{d}(t,x)\). Note that Lemmas 3.3, 3.2, and 2.1 are stability and perturbation results that can be read without knowledge on DNNs.

The remaining part of the paper is organized as follows. In Section 2 we establish a stability result on SFPEs that demonstrates the error \(E_{3}\). In Section 3 we recall basic facts on Euler-Maruyama and MLP approximations and establish the error \(E_{1}\) of the MLP approximation as well as the discretization error \(E_{2}\). Section 4 introduces a mathematical framework for DNNs and demonstrates the connection between DNNs and MLP approximations, see Lemma 4.12. Finally, Section 5 combines the results of the previous sections to prove the main results, Theorem 5.1 and Theorem 1.2.
### Notations

Let \(\left\|\cdot\right\|,\left\|\cdot\right\|\)\(\left(\cdot\right\|_{d\in\mathbb{N}}\mathbb{R}^{d})\rightarrow[0,\infty)\), \(\dim\colon(\cup_{d\in\mathbb{N}}\mathbb{R}^{d})\rightarrow\mathbb{N}\) satisfy for all \(d\in\mathbb{N}\), \(x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}\) that \(\left\|x\right\|=\sqrt{\sum_{i=1}^{d}(x_{i})^{2}}\), \(\left\|x\right\|=\max_{i\in[1,d\cap\mathbb{N}}\left|x_{i}\right|\), and \(\dim(x)=d\) and let \(\left\|\cdot\right\|_{\mathrm{F}}\colon\cup_{d\in\mathbb{N}}\mathbb{R}^{d \times d}\rightarrow[0,\infty)\) satisfy for all \(d\in\mathbb{N}\), \(x=(x_{ij})_{i,j\in[1,d]\cap\mathbb{Z}}\in\mathbb{R}^{d\times d}\) that \(\left\|x\right\|_{\mathrm{F}}^{2}=\sum_{i,j=1}^{d}\lvert x_{ij}\rvert^{2}\).

## 2. Approximation of the coefficients

In Lemma 2.1 below we approximate the solution to the SFPE (30) through solution to the SFPE (31), whose linear part, initial condition, and nonlinearity can be exactly represented through suitable DNNs.

**Lemma 2.1** (A stability result).: _Consider the notations given in Subsection 1.4, let \(T\in(0,\infty)\), \(c\in[1,\infty)\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), let \(\beta_{\varepsilon}^{d},\beta^{d}\in C(\mathbb{R}^{d},\mathbb{R}^{d})\), \(\sigma_{\varepsilon}^{d},\sigma^{d}\in C(\mathbb{R}^{d},\mathbb{R}^{d\times d})\), \(g_{\varepsilon}^{d},g^{d}\in C(\mathbb{R}^{d},\mathbb{R})\), \(f_{\varepsilon},f\in C(\mathbb{R},\mathbb{R})\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(\gamma^{d},\gamma_{\varepsilon}^{d}\colon\mathbb{R}^{2d}\rightarrow\mathbb{R}^ {d}\) be measurable, let \((\Omega,\mathcal{F},\mathbb{P},(\mathbb{F})_{t\in[0,T]})\) be a filtered probability space satisfying the usual conditions, for every \(d\in\mathbb{N}\) let \(W^{d}\colon\Omega\times[0,T]\rightarrow\mathbb{R}^{d}\), be standard \((\mathbb{F}_{t})_{t\in[0,T]}\)-Brownian motion, for every \(d\in\mathbb{N}\) let \(N^{d}\) be an \((\mathbb{F}_{t})_{t\in[0,T]}\)-Poisson random measure on \([0,\infty)\times(\mathbb{R}^{d}\setminus\{0\})\) with intensity \(\nu^{d}\), for every \(d\in\mathbb{N}\) let \(\tilde{N}^{d}(dt,dz)=N^{d}(dt,dz)-dt\,\nu^{d}(dz)\), assume for all \(d\in\mathbb{N}\) that \(\mathcal{F}_{0}\), \(N^{d}\) and \(W^{d}\) are independent, assume for all \(d\in\mathbb{N}\), \(x,y\in\mathbb{R}^{d}\), \(w_{1},w_{2}\in\mathbb{R}\), \(\varepsilon\in(0,1)\) that_

\[\left\|\beta_{\varepsilon}^{d}(x)-\beta_{\varepsilon}^{d}(y)\right\|^{2}+ \left\|\sigma_{\varepsilon}^{d}(x)-\sigma_{\varepsilon}^{d}(y)\right\|_{ \mathrm{F}}^{2}+\int_{\mathbb{R}^{d}\setminus\{0\}}\left\|\gamma_{ \varepsilon}^{d}(x,z)-\gamma_{\varepsilon}^{d}(y,z)\right\|^{2}\nu^{d}(dz) \leq c\|x-y\|^{2}, \tag{23}\]

\[\left|f_{\varepsilon}(w_{1})-f_{\varepsilon}(w_{2})\right|^{2}\leq c|w_{1}-w_ {2}|^{2},\quad T\left|g_{\varepsilon}^{d}(x)-g_{\varepsilon}^{d}(y)\right|^{2} \leq cd^{c}\|x-y\|^{2}, \tag{24}\]

\[\left\|\beta_{\varepsilon}^{d}(0)\right\|^{2}+\left\|\sigma_{ \varepsilon}^{d}(0)\right\|_{\mathrm{F}}^{2}+\int_{\mathbb{R}^{d}\setminus\{0 \}}\left\|\gamma_{\varepsilon}^{d}(0,z)\right\|^{2}\nu^{d}(dz)+T^{3}|f_{ \varepsilon}(0)|^{2}+T|g_{\varepsilon}^{d}(0)|^{2}\leq cd^{c}, \tag{25}\]

_and_

\[\left\|\beta_{\varepsilon}^{d}(x)-\beta^{d}(x)\right\|^{2}+\left\| \sigma_{\varepsilon}^{d}(x)-\sigma^{d}(x)\right\|_{\mathrm{F}}^{2}+\int_{ \mathbb{R}^{d}\setminus\{0\}}\left\|\gamma_{\varepsilon}^{d}(x,z)-\gamma^{d }(x,z)\right\|^{2}\nu^{d}(dz)\] \[\quad+\left|g_{\varepsilon}^{d}(x)-g^{d}(x)\right|^{2}+|f_{ \varepsilon}(w_{1})-f(w_{1})|^{2} \tag{26}\] \[\leq ccd^{c}(d^{c}+\|x\|^{2})+\varepsilon|w_{1}|^{4},\]

_for every \(d\in\mathbb{N}\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) let \((X_{s}^{d,t,x})_{s\in[t,T]}\), \((X_{s}^{d,\varepsilon,t,x})_{s\in[t,T]}\) be \((\mathbb{F}_{t})_{t\in[0,T]}\)-adapted cadlag processes which satisfy for all \(s\in[t,T]\) that \(\mathbb{P}\)-a.s._

\[X_{s}^{d,t,x}=x+\int_{t}^{s}\beta^{d}(X_{r-}^{d,t,x})dr+\int_{t}^{s}\sigma^{d} (X_{r-}^{d,t,x})dW_{r}^{d}+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}} \gamma^{d}(X_{r-}^{d,t,x},z)\tilde{N}^{d}(dr,dz) \tag{27}\]

_and_

\[X_{s}^{d,\varepsilon,t,x}=x+\int_{t}^{s}\beta_{\varepsilon}^{d}(X_{r-}^{d, \varepsilon,t,x})dr+\int_{t}^{s}\sigma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon, t,x})dW_{r}^{d}+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\gamma_{ \varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)\tilde{N}^{d}(dr,dz), \tag{28}\]

_and for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(u^{d},u^{d,\varepsilon}\colon[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}\) be measurable and satisfy for all \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) that \(\mathbb{E}\!\left[\left|g^{d}(X_{T}^{d,t,x})\right|\right]+\int_{t}^{T}\mathbb{E} \!\left[\left|f(u^{d}(s,X_{s}^{d,t,x}))\right|\right]ds<\infty\), \(\mathbb{E}\!\left[\left|g_{\varepsilon}^{d}(X_{T}^{d,\varepsilon,t,x})\right| \right]+\int_{t}^{T}\mathbb{E}\!\left[\left|f_{\varepsilon}(u^{d,\varepsilon}( s,X_{s}^{d,\varepsilon,t,x}))\right|\right]ds<\infty\),_

\[\sup_{s\in[0,T]}\sup_{y\in\mathbb{R}^{d}}\frac{\left|u^{d}(s,y) \right|+\left|u^{d,\varepsilon}(s,y)\right|}{1+\left\|y\right\|}<\infty, \tag{29}\]

\[u^{d}(t,x)=\mathbb{E}\!\left[g^{d}(X_{T}^{d,t,x})\right]+\int_{t}^{T}\mathbb{E} \!\left[f(u^{d}(s,X_{s}^{d,t,x}))\right]ds, \tag{30}\]

_and_

\[u^{d,\varepsilon}(t,x)=\mathbb{E}\!\left[g_{\varepsilon}^{d}(X_{T}^{d, \varepsilon,t,x})\right]+\int_{t}^{T}\mathbb{E}\!\left[f_{\varepsilon}(u^{d, \varepsilon}(s,X_{s}^{d,\varepsilon,t,x}))\right]ds. \tag{31}\]

_Then_
1. _for all_ \(d\in\mathbb{N}\)_,_ \(\varepsilon\in(0,1)\)_,_ \(t\in[0,T]\)_,_ \(x\in\mathbb{R}^{d}\) _we have that_ \[\max\biggl{\{}\mathbb{E}\biggl{[}d^{c}+\left\|X_{s}^{d,\varepsilon,t,x}\right\|^ {2}\biggr{]},\mathbb{E}\biggl{[}d^{c}+\left\|X_{s}^{d,t,x}\right\|^{2}\biggr{]} \biggr{\}}\leq(d^{c}+\|x\|^{2})e^{7c(s-t)},\] (32)
2. _for all_ \(d\in\mathbb{N}\)_,_ \(\varepsilon\in(0,1)\)_,_ \(t\in[0,T]\)_,_ \(x,y\in\mathbb{R}^{d}\) _we have that_ \[\left|u^{d,\varepsilon}(t,x)-u^{d,\varepsilon}(t,y)\right|\leq 2(cd^{c}T^{-1})^{ \frac{1}{2}}\|x-y\|e^{5cT+2cT^{2}},\] (33) _and_
3. _for all_ \(d\in\mathbb{N}\)_,_ \(\varepsilon\in(0,1)\)_,_ \(t\in[0,T]\)_,_ \(x\in\mathbb{R}^{d}\) _we have that_ \[\left|u^{d,\varepsilon}(t,x)-u^{d}(t,x)\right|\leq 2cd^{c}\varepsilon^{ \frac{1}{2}}(d^{c}+\|x\|^{2})e^{24cT+5cT^{2}}.\] (34)

Proof of Lemma 2.1.: Throughout this proof let \(\langle\cdot,\cdot\rangle\colon\cup_{d\in\mathbb{R}}\mathbb{R}^{d}\times \mathbb{R}^{d}\to\mathbb{R}\) satisfy for every \(d\in\mathbb{N}\), \(x=(x_{1},\ldots,x_{d}),y=(y_{1},\ldots,y_{d})\in\mathbb{R}^{d}\) that \(\langle x,y\rangle=\sum_{i=1}^{d}x_{i}y_{i}\). First, the fact that \(\forall\,d\in\mathbb{N},x,y\in\mathbb{R}^{d}\colon\|x+y\|^{2}\leq 2\|x\|^{2}+2 \|y\|^{2}\), (25), and (23) show for all \(d\in\mathbb{N}\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\|\beta_{\varepsilon}^{d}(x)\|^{2}\leq 2\|\beta_{\varepsilon}^{d}(0)\|^{2}+2\| \beta_{\varepsilon}^{d}(x)-\beta_{\varepsilon}^{d}(0)\|^{2}\leq 2cd^{c}+2c\|x\|^{2} =2c(d^{c}+\|x\|^{2}). \tag{35}\]

Similarly, we have for all \(d\in\mathbb{N}\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\left\|\sigma_{\varepsilon}^{d}(x)\right\|_{\mathrm{F}}^{2}\leq 2c(d^{c}+\|x\|^{2}). \tag{36}\]

Next, the fact that \(\forall\,d\in\mathbb{N},x,y\in\mathbb{R}^{d}\colon\|x+y\|^{2}\leq 2\|x\|^{2}+2 \|y\|^{2}\), (25), and (23) show for all \(d\in\mathbb{N}\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\int_{\mathbb{R}^{d}\setminus\{0\}}\left\|\gamma_{ \varepsilon}^{d}(x,z)\right\|^{2}\nu^{d}(dz)&\leq\int_{\mathbb{R }^{d}\setminus\{0\}}2\left\|\gamma_{\varepsilon}^{d}(0,z)\right\|^{2}+2 \left\|\gamma_{\varepsilon}^{d}(x,z)-\gamma_{\varepsilon}^{d}(0,z)\right\|^{2 }\nu^{d}(dz)\\ &\leq 2cd^{c}+2c\|x^{2}\|=2c(d^{c}+\|x\|^{2}).\end{split} \tag{37}\]

Next, Ito's formula (see, e.g., [20, Theorem 3.1]) and (28) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that \(\mathbb{P}\)-a.s. we have that

\[\begin{split}\left\|X_{s}^{d,\varepsilon,t,x}\right\|^{2}& =\|x\|^{2}+\int_{t}^{s}\left(2\left\langle X_{r}^{d,\varepsilon,t, x},\beta_{\varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x})\right\rangle+\left\| \sigma_{\varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x})\right\|_{\mathrm{F}}^{2} \right)dr\\ &\quad+2\int_{t}^{s}\sum_{i,j=1}^{d}\left(X_{r-}^{d,\varepsilon,t,x}\right)_{i}\left(\sigma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x}) \right)_{ij}d(W_{j}^{d})_{r}\\ &\quad+2\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\left\langle X _{r-}^{d,\varepsilon,t,x},\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z )\right\rangle\tilde{N}^{d}(dz,dr)\\ &\quad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\left\| \gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)\right\|^{2}N^{d}(dz,dr ).\end{split} \tag{38}\]

Next, for every \(d,n\in\mathbb{N}\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) let \(\tau_{n}^{d,\varepsilon,x}\colon\Omega\to\mathbb{R}\) satisfy that

\[\begin{split}\tau_{n}^{d,\varepsilon,x}=\inf\biggl{\{}& s\in[t,T]\colon\int_{t}^{s}\left(2\left\langle X_{r}^{d,\varepsilon,t,x},\beta_{ \varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x})\right\rangle+\left\|\sigma_{ \varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x})\right\|_{\mathrm{F}}^{2}\right)dr \\ &\quad+\int_{t}^{s}\sum_{i,j=1}^{d}\left|\left(X_{r}^{d, \varepsilon,t,x}\right)_{i}\left(\sigma_{\varepsilon}^{d}(X_{r}^{d,\varepsilon,t, x})\right)_{ij}\right|^{2}dr\\ &\quad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\sum_{i=1}^{d} \left|\left(X_{r}^{d,\varepsilon,t,x}\right)_{i}\left(\gamma_{\varepsilon}^{d}(X_{ r}^{d,\varepsilon,t,x},z)\right)_{i}\right|^{2}\nu^{d}(dz)\,dr\\ &\quad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\left\| \gamma_{\varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x},z)\right\|_{\mathrm{F}}^{2} \nu^{d}(dz)\,dr\geq n\biggr{\}\wedge T\end{split} \tag{39}\]
(with the convention that \(\inf\emptyset=\infty\)). Then (38), the fact that \(\forall\,d\in\mathbb{N},x,y\in\mathbb{R}^{d}\colon 2\langle x,y\rangle\leq\|x\|^{2}+\|y\|^ {2}\), (35), (36), (37), and the fact that \(c\geq 1\) show for all \(d,n\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\bigg{[}d^{c}+\left\|X_{s\wedge\tau_{n}^{d, \varepsilon,x}}^{d,\varepsilon,t,x}\right\|^{2}\bigg{]}\\ &=d^{c}+\|x\|^{2}+\mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{n}^{d, \varepsilon,x}}\left(2\left\langle X_{r}^{d,\varepsilon,t,x},\beta_{ \varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x})\right\rangle+\left\|\sigma_{ \varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x})\right\|_{\mathrm{F}}^{2}\right)dr \Bigg{]}\\ &\quad+\mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{n}^{d, \varepsilon,x}}\int_{\mathrm{R}^{d}\setminus\{0\}}\left\|\gamma_{\varepsilon }^{d}(X_{r}^{d,\varepsilon,t,x},z)\right\|^{2}\nu^{d}(dz)\,dr\Bigg{]}\\ &\leq d^{c}+\|x\|^{2}+\mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{ n}^{d,\varepsilon,x}}\left\|X_{r}^{d,\varepsilon,t,x}\right\|^{2}dr\Bigg{]}+ \mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{n}^{d,\varepsilon,x}}\left\|\beta_{ \varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x})\right\|^{2}dr\Bigg{]}\\ &\quad+\mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{n}^{d, \varepsilon,x}}\left\|\sigma_{\varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x}) \right\|_{\mathrm{F}}^{2}dr\Bigg{]}+\mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{ n}^{d,\varepsilon,x}}\int_{\mathrm{R}^{d}\setminus\{0\}}\left\|\gamma_{ \varepsilon}^{d}(X_{r}^{d,\varepsilon,t,x},z)\right\|^{2}\nu^{d}(dz)\,dr \Bigg{]}\end{split} \tag{40}\]

and

\[\begin{split}&\mathbb{E}\bigg{[}d^{c}+\left\|X_{s\wedge\tau_{n}^{ d,\varepsilon,x}}^{d,\varepsilon,t,x}\right\|^{2}\bigg{]}\\ &\leq d^{c}+\|x\|^{2}+\mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{ n}^{d,\varepsilon,x}}\left(d^{c}+\left\|X_{r}^{d,\varepsilon,t,x}\right\|^{2} \right)dr\Bigg{]}+\mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{n}^{d,\varepsilon, x}}2c\left(d^{c}+\left\|X_{r}^{d,\varepsilon,t,x}\right\|^{2}\right)dr \Bigg{]}\\ &\quad+\mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{n}^{d, \varepsilon,x}}2c\left(d^{c}+\left\|X_{r}^{d,\varepsilon,t,x}\right\|_{ \mathrm{F}}^{2}\right)dr\Bigg{]}+\mathbb{E}\Bigg{[}\int_{t}^{s\wedge\tau_{n}^ {d,\varepsilon,x}}2c\left(d^{c}+\left\|X_{r}^{d,\varepsilon,t,x}\right\|^{2} \right)dr\Bigg{]}\\ &\leq d^{c}+\|x\|^{2}+7c\int_{t}^{s}\mathbb{E}\left[d^{c}+\left\| X_{r\wedge\tau_{n}^{d,\varepsilon,x}}^{d,\varepsilon,t,x}\right\|^{2}dr\right]. \end{split} \tag{41}\]

This, Fatou's lemma, and Gronwall's inequality show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\mathbb{E}\bigg{[}d^{c}+\left\|X_{s}^{d,\varepsilon,t,x}\right\|^{2}\bigg{]} \leq\liminf_{n\to\infty}\mathbb{E}\bigg{[}d^{c}+\left\|X_{s\wedge\tau_{n}^{d,\varepsilon,x}}^{d,\varepsilon,t,x}\right\|^{2}\bigg{]}\leq(d^{c}+\|x\|^{2}) e^{7c(s-t)}. \tag{42}\]

Next, using (26) and letting \(\varepsilon\) tend to zero in (23) and (25) we obtain that

\[\left\|\beta^{d}(x)-\beta^{d}(y)\right\|^{2}+\left\|\sigma^{d}(x)-\sigma^{d}( y)\right\|^{2}+\int_{\mathrm{R}^{d}\setminus\{0\}}\left\|\gamma^{d}(x,z)- \gamma^{d}(y,z)\right)\right\|^{2}\nu^{d}(dz)\,dr\leq c\|x-y\|^{2} \tag{43}\]

and

\[\left\|\beta^{d}(0)\right\|^{2}+\left\|\sigma^{d}(0)\right\|^{2}+\int_{ \mathrm{R}^{d}\setminus\{0\}}\left\|\gamma^{d}(0,z)\right\|^{2}\nu^{d}(dz) \leq cd^{c}. \tag{44}\]

Using a similar argument as that for (42) we then obtain for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\) that

\[\mathbb{E}\bigg{[}d^{c}+\left\|X_{s}^{d,t,x}\right\|^{2}\bigg{]}\leq(d^{c}+\|x \|^{2})e^{7c(s-t)}. \tag{45}\]

This and (42) show (i).

Next, (26) and (45) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\bigg{[}\int_{t}^{s}\left\|\beta_{ \varepsilon}^{d}(X_{r-}^{d,t,x})-\beta^{d}(X_{r-}^{d,t,x})\right\|^{2}dr \bigg{]}\\ &\leq\int_{t}^{s}\varepsilon cd^{c}\left(d^{c}+\mathbb{E}\left[ \left\|X_{r}^{d,t,x}\right\|^{2}\right]\right)dr\\ &\leq\int_{t}^{s}\varepsilon cd^{c}(d^{c}+\|x\|^{2})e^{7c(r-t)} \,dr\leq\varepsilon cd^{c}(d^{c}+\|x\|^{2})(s-t)e^{7cT}\end{split} \tag{46}\]

and similarly

\[\int_{t}^{s}\mathbb{E}\bigg{[}\left\|\sigma_{\varepsilon}^{d}(X_{r-}^{d,t,x})- \sigma^{d}(X_{r-}^{d,t,x})\right\|_{\mathrm{F}}^{2}\bigg{]}\,dr\leq ccd^{c}(d^{c }+\|x\|^{2})(s-t)e^{7cT}. \tag{47}\]
Next, (26) and (45) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}} \mathbb{E}\left[\left\|\gamma_{\varepsilon}^{d}(X_{r-}^{d,t,x},z)-\gamma^{d}(X _{r-}^{d,t,x},z)\right\|^{2}\right]\nu^{d}(dz)\,dr\\ &\leq\int_{t}^{s}\varepsilon cd^{c}\left(d^{c}+\mathbb{E}\left[ \left\|X_{r}^{d,t,x}\right\|^{2}\right]\right)dr\\ &\leq\int_{t}^{s}\varepsilon cd^{c}(d^{c}+\|x\|^{2})e^{\tau c(r-t )}\,dr\leq\varepsilon cd^{c}(d^{c}+\|x\|^{2})(s-t)e^{\tau cT}.\end{split} \tag{48}\]

Next, Holder's inequality, the fact that \(\forall\,d\in\mathbb{N},x,y\in\mathbb{R}^{d}\colon\|x+y\|^{2}\leq 2\|x\|^{2}+2\|y\|^ {2}\), (23), and (46) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\left[\left\|\int_{t}^{s}(\beta_{ \varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x})-\beta^{d}(X_{r-}^{d,t,x}))\,dr \right\|^{2}\right]\\ &\leq\mathbb{E}\Bigg{[}\left(\int_{t}^{s}\left\|\beta_{ \varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x})-\beta^{d}(X_{r-}^{d,t,x})\right\| dr\right)^{2}\Bigg{]}\\ &\leq\mathbb{E}\Bigg{[}\left(\int_{t}^{s}dr\right)\left(\int_{t} ^{s}\left\|\beta_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x})-\beta^{d}(X_{r- }^{d,t,x})\right\|^{2}dr\right)\Bigg{]}\\ &\leq T\mathbb{E}\Bigg{[}\int_{t}^{s}\left\|\beta_{\varepsilon} ^{d}(X_{r-}^{d,\varepsilon,t,x})-\beta^{d}(X_{r-}^{d,t,x})\right\|^{2}dr \Bigg{]}\\ &\leq 2T\mathbb{E}\Bigg{[}\int_{t}^{s}\left\|\beta_{\varepsilon} ^{d}(X_{r-}^{d,\varepsilon,t,x})-\beta_{\varepsilon}^{d}(X_{r-}^{d,t,x})\right\| ^{2}dr\Bigg{]}+2T\mathbb{E}\left[\int_{t}^{s}\left\|\beta_{\varepsilon}^{d}(X _{r-}^{d,t,x})-\beta^{d}(X_{r-}^{d,t,x})\right\|^{2}dr\right]\\ &\leq 2Tc\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,\varepsilon,t,x }-X_{r}^{d,t,x}\right\|^{2}\right]dr+2T\cdot\varepsilon cd^{c}(d^{c}+\|x\|^{2 })(s-t)e^{\tau cT}\\ &=2cT\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,\varepsilon,t, x}-X_{r}^{d,t,x}\right\|^{2}\right]dr+2\varepsilon cd^{c}(d^{c}+\|x\|^{2})T(s-t)e^{ \tau cT}.\end{split} \tag{49}\]

Furthermore, Ito's isometry, the fact that \(\forall\,d\in\mathbb{N},x,y\in\mathbb{R}^{d\times d}\colon\|x+y\|_{\rm F}^{2} \leq 2\|x\|_{\rm F}^{2}+2\|y\|_{\rm F}^{2}\), (23), and (47) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\Bigg{[}\left\|\int_{t}^{s}(\sigma_{ \varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x})-\sigma^{d}(X_{r-}^{d,t,x}))\,dW_{r }^{d}\right\|^{2}\Bigg{]}\\ &=\mathbb{E}\Bigg{[}\int_{t}^{s}\left\|\sigma_{\varepsilon}^{d}(X _{r-}^{d,\varepsilon,t,x})-\sigma^{d}(X_{r-}^{d,t,x})\right\|_{\rm F}^{2}dr \Bigg{]}\\ &\leq 2\int_{t}^{s}\mathbb{E}\Bigg{[}\left\|\sigma_{\varepsilon}^{d}(X _{r-}^{d,\varepsilon,t,x})-\sigma_{\varepsilon}^{d}(X_{r-}^{d,t,x})\right\|_{ \rm F}^{2}\Bigg{]}dr+2\int_{t}^{s}\mathbb{E}\left[\left\|\sigma_{\varepsilon}^{ d}(X_{r-}^{d,t,x})-\sigma^{d}(X_{r-}^{d,t,x})\right\|_{\rm F}^{2}\right]dr\\ &\leq 2c\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,\varepsilon,t,x}-X_{ r}^{d,t,x}\right\|_{\rm F}^{2}\right]dr+2\varepsilon cd^{c}(d^{c}+\|x\|^{2})(s-t)e^{ \tau cT}.\end{split} \tag{50}\]

Next, Ito's isometry (see, e.g., [9, Proposition 8.8]), the fact that \(\forall\,d\in\mathbb{N},x,y\in\mathbb{R}^{d}\colon\|x+y\|^{2}\leq 2\|x\|^{2}+2\|y\|^ {2}\), (23), and (48) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\left[\left\|\int_{t}^{s}\int_{\mathbb{R}^{d} \setminus\{0\}}(\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)-\gamma^{ d}(X_{r-}^{d,t,x},z))\,\tilde{N}^{d}(dr,dz)\right\|^{2}\right]\\ &=\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\mathbb{E}\,\left[ \left\|\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)-\gamma^{d}(X_{r-}^{d,t,x},z))\right\|^{2}\right]\nu^{d}(dz)\,dr\\ &\leq 2\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\mathbb{E}\left[ \left\|\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)-\gamma_{ \varepsilon}^{d}(X_{r-}^{d,t,x},z))\right\|^{2}\nu^{d}(dz)\,dr\\ &\quad+2\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\mathbb{E} \left[\left\|\gamma_{\varepsilon}^{d}(X_{r-}^{d,t,x},z)-\gamma^{d}(X_{r-}^{d,t,x},z) \right\|^{2}\right]\nu^{d}(dz)\,dr\\ &\leq 2c\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,\varepsilon,t,x}-X_{ r}^{d,t,x}\right\|^{2}\right]dr+2\varepsilon cd^{c}(d^{c}+\|x\|^{2})(s-t)e^{\tau cT}.\end{split} \tag{51}\]
Next, (27) and (28) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that \(\mathbb{P}\)-a.s.

\[\begin{split}& X_{s}^{d,\varepsilon,t,x}-X_{s}^{d,t,x}\\ &=\int_{t}^{s}(\beta_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t, x})-\beta^{d}(X_{r-}^{d,t,x}))\,dr+\int_{t}^{s}(\sigma_{\varepsilon}^{d}(X_{r-}^{d, \varepsilon,t,x})-\sigma^{d}(X_{r-}^{d,t,x}))\,dW_{r}^{d}\\ &\quad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}(\gamma_{ \varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)-\gamma^{d}(X_{r-}^{d,t,x},z)) \,\tilde{N}^{d}(dr,dz).\end{split} \tag{52}\]

This, the fact that \(\forall\,d\in\mathbb{N},x,y,z\in\mathbb{R}^{d}\colon\|x+y+z\|^{2}\leq 3\|x\|^{2}+3 \|y\|^{2}+3\|z\|^{2}\), (49), (50), and (51) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\left[\left\|X_{s}^{d,\varepsilon,t,x}-X_{s} ^{d,t,x}\right\|^{2}\right]\\ &\leq 3\mathbb{E}\left[\left\|\int_{t}^{s}(\beta_{\varepsilon}^{d}(X _{r-}^{d,\varepsilon,t,x})-\beta^{d}(X_{r-}^{d,t,x}))dr\right\|^{2}\right]+3 \mathbb{E}\left[\left\|\int_{t}^{s}(\sigma_{\varepsilon}^{d}(X_{r-}^{d, \varepsilon,t,x})-\sigma^{d}(X_{r-}^{d,t,x}))\,dW_{r}^{d}\right\|^{2}\right] \\ &\quad+3\mathbb{E}\left[\left\|\int_{t}^{s}\int_{\mathbb{R}^{d} \setminus\{0\}}(\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)-\gamma ^{d}(X_{r-}^{d,t,x},z))\,\tilde{N}^{d}(dr,dz)\right\|^{2}\right]\\ &\leq 3\left[2cT\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d, \varepsilon,t,x}-X_{r}^{d,t,x}\right\|^{2}\right]dr+2\varepsilon cd^{c}(d^{c} +\|x\|^{2})T(s-t)e^{\tau cT}\right]\\ &\quad+3\left[2c\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d, \varepsilon,t,x}-X_{r}^{d,t,x}\right\|_{\mathbb{F}}^{2}\right]dr+2\varepsilon cd ^{c}(d^{c}+\|x\|^{2})(s-t)e^{\tau cT}\right]\\ &\quad+3\left[2c\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d, \varepsilon,t,x}-X_{r}^{d,t,x}\right\|^{2}\right]dr+2\varepsilon cd^{c}(d^{c} +\|x\|^{2})(s-t)e^{\tau cT}\right]\\ &=(12c+6cT)\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d, \varepsilon,t,x}-X_{r}^{d,t,x}\right\|^{2}\right]dr+3\left(2T+4\right) \varepsilon cd^{c}(d^{c}+\|x\|^{2})(s-t)e^{7cT}.\end{split} \tag{53}\]

This, Gronwall's inequality, (42), (45), and the fact that \(3\,(2T+4)\leq 12(T+1)\leq 12e^{cT}\) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\mathbb{E}\left[\left\|X_{s}^{d,\varepsilon,t,x}-X_{s }^{d,t,x}\right\|^{2}\right]&\leq 3\,(2T+4)\,\varepsilon cd^{c}(d^{c} +\|x\|^{2})(s-t)e^{\tau cT}e^{(12c+6cT)T}\\ &\leq 12e^{cT}\varepsilon cd^{c}(d^{c}+\|x\|^{2})e^{\tau cT}e^{(12c+6 cT)T}(s-t)\\ &=12\varepsilon cd^{c}(d^{c}+\|x\|^{2})e^{20cT+6cT^{2}}(s-t). \end{split} \tag{54}\]

Next, (28), the fact that \(\forall\,d\in\mathbb{N},x_{1},x_{2},x_{3},x_{4}\in\mathbb{R}^{d}\colon\| \sum_{i=1}^{4}x_{i}\|^{2}\leq 4\sum_{i=1}^{4}\|x_{i}\|^{2}\), Jensen's inequality, Ito's isometry, and (23) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\mathbb{E}\left[\left\|X_{s}^{d,\varepsilon,t,x}-X_{s }^{d,\varepsilon,t,y}\right\|^{2}\right]&\leq 4\|x-y\|^{2}+4\mathbb{E}\left[\left\|\int_{t}^{s}( \beta_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x})-\beta_{\varepsilon}^{d}(X_ {r-}^{d,\varepsilon,t,y}))dr\right\|^{2}\right]\\ &\quad+4\mathbb{E}\left[\left\|\int_{t}^{s}(\sigma_{\varepsilon}^{ d}(X_{r-}^{d,\varepsilon,t,x})-\sigma_{\varepsilon}^{d}(X_{r-}^{d, \varepsilon,t,y}))dW_{r}^{d}\right\|^{2}\right]\\ &\quad+4\mathbb{E}\left[\left\|\int_{t}^{s}\int_{\mathbb{R}^{d} \setminus\{0\}}(\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)-\gamma _{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,y},z))\tilde{N}^{d}(dr,dz)\right\| ^{2}\right]\end{split} \tag{55}\]
and

\[\mathbb{E}\left[\left\|X_{s}^{d,\varepsilon,t,x}-X_{s}^{d, \varepsilon,t,y}\right\|^{2}\right] \tag{56}\] \[\leq 4\|x-y\|^{2}+4T\mathbb{E}\left[\int_{t}^{s}\left\|\beta_{ \varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x})-\beta_{\varepsilon}^{d}(X_{r-}^{ d,\varepsilon,t,y})\right\|^{2}dr\right]\] \[\quad+4\mathbb{E}\left[\int_{t}^{s}\left\|\sigma_{\varepsilon}^{d }(X_{r-}^{d,\varepsilon,t,x})-\sigma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon, t,y})\right\|^{2}_{\text{F}}dr\right]\] \[\quad+4\mathbb{E}\left[\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus \{0\}}\left\|\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)-\gamma_{ \varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,y},z)\right\|^{2}\nu^{d}(dz)\,dr\right]\] \[\leq 4\|x-y\|^{2}+4T\mathbb{E}\left[\int_{t}^{s}c\left\|X_{r-}^{ d,\varepsilon,t,x}-X_{r-}^{d,\varepsilon,t,y}\right\|^{2}dr\right]+4\mathbb{E} \left[\int_{t}^{s}c\left\|X_{r-}^{d,\varepsilon,t,x}-X_{r-}^{d,\varepsilon,t,y}\right\|^{2}dr\right]\] \[\quad+4\mathbb{E}\left[\int_{t}^{s}c\left\|X_{r-}^{d,\varepsilon,t,x}-X_{r-}^{d,\varepsilon,t,y}\right\|^{2}dr\right]\] \[=4\|x-y\|^{2}+(4Tc+8c)\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{ d,\varepsilon,t,x}-X_{r}^{d,\varepsilon,t,y}\right\|^{2}\right]dr.\]

This, Gronwall's lemma, and (42) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\mathbb{E}\left[\left\|X_{s}^{d,\varepsilon,t,x}-X_{s}^{d, \varepsilon,t,y}\right\|^{2}\right]\leq 4\|x-y\|^{2}e^{(4Tc+8c)(s-t)}\leq 4\|x-y\|^{2}e ^{(4Tc+8c)T}=4\|x-y\|^{2}e^{8cT+4cT^{2}}. \tag{57}\]

Next, (24), Jensen's inequality, and (54) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) that

\[\mathbb{E}\left[\left|g_{\varepsilon}^{d}(X_{T}^{d,\varepsilon, t,x})-g_{\varepsilon}^{d}(X_{T}^{d,t,x})\right|\right] \leq\mathbb{E}\left[(cd^{c})^{\frac{1}{2}}T^{-\frac{1}{2}}\left\| X_{T}^{d,\varepsilon,t,x}-X_{T}^{d,t,x}\right\|\right] \tag{58}\] \[\leq(cd^{c})^{\frac{1}{2}}T^{-\frac{1}{2}}\left(\mathbb{E}\left[ \left\|X_{T}^{d,\varepsilon,t,x}-X_{T}^{d,t,x}\right\|^{2}\right]\right)^{ \frac{1}{2}}\] \[\leq(cd^{c})^{\frac{1}{2}}T^{-\frac{1}{2}}\left(12\varepsilon d ^{c}(d^{c}+\|x\|^{2})e^{20cT+6cT^{2}}T\right)^{\frac{1}{2}}\] \[\leq 4(\varepsilon cd^{2c})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{ \frac{1}{2}}e^{10cT+3cT^{2}}.\]

Next, (26), Jensen's inequality, and (45) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) that

\[\mathbb{E}\left[\left|g_{\varepsilon}^{d}(X_{T}^{d,t,x})-g^{d}(X_ {T}^{d,t,x})\right|\right] \leq\mathbb{E}\left[\left|\varepsilon cd^{c}\left(d^{c}+\left\|X _{T}^{d,t,x}\right\|^{2}\right)\right|^{\frac{1}{2}}\right] \leq(\varepsilon cd^{c})^{\frac{1}{2}}\left(\mathbb{E}\left[d^{c}+ \left\|X_{T}^{d,t,x}\right\|^{2}\right]\right)^{\frac{1}{2}} \tag{59}\] \[\leq(\varepsilon cd^{c})^{\frac{1}{2}}\left((d^{c}+\|x\|^{2})^{ \frac{1}{2}}e^{7cT}\right)^{\frac{1}{2}}\] \[=(\varepsilon cd^{c})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2 }}e^{3.5cT}.\]

This, the triangle inequality, and (58) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) that

\[\mathbb{E}\left[\left|g_{\varepsilon}^{d}(X_{T}^{d,\varepsilon,t,x })-g^{d}(X_{T}^{d,t,x})\right|\right] \leq\mathbb{E}\left[\left|g_{\varepsilon}^{d}(X_{T}^{d, \varepsilon,t,x})-g_{\varepsilon}^{d}(X_{T}^{d,t,x})\right|\right]+\mathbb{E} \left[\left|g_{\varepsilon}^{d}(X_{T}^{d,t,x})-g^{d}(X_{T}^{d,t,x})\right|\right] \tag{60}\] \[\leq 4(\varepsilon cd^{2c})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{ \frac{1}{2}}e^{10cT+3cT^{2}}+(\varepsilon cd^{c})^{\frac{1}{2}}(d^{c}+\|x\|^{2 })^{\frac{1}{2}}e^{3.5cT}\] \[\leq 5(\varepsilon cd^{2c})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{ \frac{1}{2}}e^{10cT+3cT^{2}}.\]
This, Jensen's inequality, and (45) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(s\in[0,T]\), \(t\in[s,T]\), \(x\in\mathds{R}^{d}\) that

\[\begin{split}\mathbb{E}\left[\mathbb{E}\Big{[}\Big{|}g_{ \varepsilon}^{d}(X_{T}^{d,\varepsilon,t,\tilde{x}})-g^{d}(X_{T}^{d,t,\tilde{x} })\Big{|}\Big{]}\Big{|}\Big{|}_{\tilde{x}=X_{t}^{d,s,x}}\right]& \leq 5(\varepsilon cd^{2c})^{\frac{1}{2}}\mathbb{E}\left[ \left(d^{c}+\left\|X_{t}^{d,s,x}\right\|^{2}\right)^{\frac{1}{2}}\right]e^{10cT +3cT^{2}}\\ &\leq 5(\varepsilon cd^{2c})^{\frac{1}{2}}\left(\mathbb{E}\Big{[}d^{ c}+\left\|X_{t}^{d,s,x}\right\|^{2}\right)\right]^{\frac{1}{2}}e^{10cT+3cT^{2}}\\ &\leq 5(\varepsilon cd^{2c})^{\frac{1}{2}}\left((d^{c}+\|x\|^{2})^{ \frac{1}{2}}e^{10cT+3cT^{2}}\right.\\ &\leq 5(\varepsilon cd^{2c})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{ \frac{1}{2}}e^{14cT+3cT^{2}}.\end{split} \tag{61}\]

Next, the triangle inequality, (25), and (24) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(x\in\mathds{R}^{d}\), \(w\in\mathds{R}\) that

\[\begin{split}\Big{|}g_{\varepsilon}^{d}(x)\Big{|}\leq\Big{|}g_{ \varepsilon}^{d}(0)\Big{|}+\Big{|}g_{\varepsilon}^{d}(x)-g_{\varepsilon}^{d}( 0)\Big{|}&\leq(cd^{c}T^{-1})^{\frac{1}{2}}+(cd^{c}T^{-1})^{ \frac{1}{2}}\|x\|\\ &\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}} \end{split} \tag{62}\]

and

\[|f_{\varepsilon}(w)|\leq|f_{\varepsilon}(0)|+|f_{\varepsilon}(w)-f_{ \varepsilon}(0)|\leq(cd^{c}T^{-3})^{\frac{1}{2}}+c^{\frac{1}{2}}|w|. \tag{63}\]

This and (26) show for all \(d\in\mathbb{N}\), \(x\in\mathds{R}^{d}\) that

\[|g^{d}(x)|=\lim_{\varepsilon\to 0}\Big{|}g_{\varepsilon}^{d}(x)\Big{|}\leq 2(cd^{ c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}. \tag{64}\]

This and (45) show for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(x\in\mathds{R}^{d}\) that

\[\begin{split}\left\|g^{d}(X_{T}^{d,t,x})\right\|_{L^{2}(\mathrm{P} )}&\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\left\|\left(d^{c}+\left\|X_{T}^{ d,t,x}\right\|^{2}\right)^{\frac{1}{2}}\right\|_{L^{2}(\mathrm{P})}\\ &\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\left(\mathbb{E}\left[d^{c}+\left\|X _{T}^{d,t,x}\right\|^{2}\right]\right)^{\frac{1}{2}}\\ &\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\left((d^{c}+\|x\|^{2})^{c} ^{\varepsilon TcT}\right)^{\frac{1}{2}}\\ &=2(cd^{c}T^{-1})^{\frac{1}{2}}\left(d^{c}+\|x\|^{2}\right)^{ \frac{1}{2}}e^{3.5cT}.\end{split} \tag{65}\]

This, (31), the triangle inequality, the disintegration theorem, the flow property, and (63) show for all \(d\in\mathbb{N}\), \(s\in[0,T]\), \(t\in[s,T]\), \(x\in\mathds{R}^{d}\) that

(66)

This, Gronwall's lemma, (29), (45), and the fact that \(c\in[1,\infty)\) show for all \(d\in\mathbb{N}\), \(s\in[0,T]\), \(t\in[s,T]\), \(x\in\mathds{R}^{d}\) that

\[\left\|u^{d}(t,X_{t}^{d,s,x})\right\|_{L^{2}(\mathrm{P})}\leq 3(cd^{c}T^{-1})^{ \frac{1}{2}}\left(d^{c}+\|x\|^{2}\right)^{\frac{1}{2}}e^{3.5cT}e^{c^{1/2}T} \leq 3(cd^{c}T^{-1})^{\frac{1}{2}}\left(d^{c}+\|x\|^{2}\right)^{\frac{1}{2} }e^{4.5cT}. \tag{67}\]
Hence, for all \(d\in\mathbb{N}\), \(s\in[0,T]\), \(t\in[s,T]\), \(x\in\mathbb{R}^{d}\) we have that

\[|u^{d}(t,x)|\leq 3(cd^{c}T^{-1})^{\frac{1}{2}}\left(d^{c}+\|x\|^{2}\right)^{ \frac{1}{2}}e^{4.5cT}. \tag{68}\]

Next, (31) and the triangle inequality show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x,y\in\mathds{R}^{d}\) that

\[\begin{split}\Big{|}u^{d,\varepsilon}(t,x)-u^{d,\varepsilon}(t, y)\Big{|}&\leq\mathbb{E}\Big{[}\Big{|}g_{\varepsilon}^{d}(X_{T}^{d, \varepsilon,t,x})-g_{\varepsilon}^{d}(X_{T}^{d,\varepsilon,t,y})\Big{|} \Big{]}\\ &\quad+\int_{t}^{T}\mathbb{E}\Big{[}\Big{|}f_{\varepsilon}(u^{d, \varepsilon}(X_{r}^{d,\varepsilon,t,x}))-f_{\varepsilon}(u^{d,\varepsilon}(X_ {r}^{d,\varepsilon,t,y}))\Big{|}\Big{]}\,dr.\end{split} \tag{69}\]

This, the triangle inequality, the disintegration theorem, (24), Jensen's inequality, the fact that \(c\in[1,\infty)\), and (57) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(s\in[0,T]\), \(t\in[s,T]\), \(x,y\in\mathds{R}^{d}\) that

(70)

This, Gronwall's inequality, (29), and (42) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(s\in[0,T]\), \(t\in[s,T]\), \(x,y\in\mathds{R}^{d}\) that

\[\begin{split}\mathbb{E}\Big{[}\Big{|}u^{d,\varepsilon}(t,X_{t}^{d,\varepsilon,s,x})-u^{d,\varepsilon}(t,X_{t}^{d,\varepsilon,s,y})\Big{|}\Big{]} &\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\|x-y\|e^{4cT+2cT^{2}}\cdot e^{cT}\\ &=2(cd^{c}T^{-1})^{\frac{1}{2}}\|x-y\|e^{5cT+2cT^{2}}\end{split} \tag{71}\]

and hence

\[\begin{split}\Big{|}u^{d,\varepsilon}(t,x)-u^{d,\varepsilon}(t, y)\Big{|}\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\|x-y\|e^{5cT+2cT^{2}}.\end{split} \tag{72}\]

This shows (ii).

Next, (24), (72), Jensen's inequality, and (54) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathds{R}^{d}\) that

(73)
This, the triangle inequality, (24), (26), the fact that \(c\in[1,\infty)\), the fact that \(cT\leq e^{cT}\), and (68) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) that

\[\begin{split}&\int_{t}^{T}\mathbb{E}\Big{[}\Big{|}f_{\varepsilon}(u ^{d,\varepsilon}(s,X_{s}^{d,\varepsilon,t,x}))-f(u^{d}(s,X_{s}^{d,t,x}))\Big{|} \Big{]}\,ds\\ &\leq T\sup_{s\in[t,T]}\mathbb{E}\Big{[}\Big{|}f_{\varepsilon}(u ^{d,\varepsilon}(s,X_{s}^{d,\varepsilon,t,x}))-f_{\varepsilon}(u^{d, \varepsilon}(s,X_{s}^{d,t,x}))\Big{|}\Big{]}\\ &\quad+\int_{t}^{T}\mathbb{E}\Big{[}\Big{|}f_{\varepsilon}(u^{d, \varepsilon}(s,X_{s}^{d,t,x}))-f_{\varepsilon}(u^{d}(s,X_{s}^{d,t,x}))\Big{|} \Big{]}\,ds\\ &\quad+T\sup_{s\in[t,T]}\mathbb{E}\Big{[}\Big{|}f_{\varepsilon} \big{(}u^{d}(s,X_{s}^{d,t,x}))-f(u^{d}(s,X_{s}^{d,t,x}))\Big{|}\Big{]}\\ &\leq T\cdot 8c^{\frac{3}{2}}d^{c}\varepsilon^{\frac{1}{2}}(d^{c}+\|x \|^{2})^{\frac{1}{2}}e^{15cT+5cT^{2}}+\int_{t}^{T}c^{\frac{1}{2}}\mathbb{E} \Big{[}\Big{|}u^{d,\varepsilon}(s,X_{s}^{d,t,x})-u^{d}(s,X_{s}^{d,t,x})\Big{|} \Big{]}\,ds\\ &\quad+T\sup_{s\in[t,T]}\Big{[}\varepsilon^{\frac{1}{2}}\mathbb{ E}\Big{[}\Big{|}u^{d}(s,X_{s}^{d,t,x})\Big{|}^{2}\Big{]}\Big{]}\end{split} \tag{74}\]

and

\[\begin{split}&\int_{t}^{T}\mathbb{E}\Big{[}\Big{|}f_{\varepsilon} (u^{d,\varepsilon}(s,X_{s}^{d,\varepsilon,t,x}))-f(u^{d}(s,X_{s}^{d,t,x})) \Big{|}\Big{]}\,ds\\ &\leq 8cd^{c}\varepsilon^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{ \frac{1}{2}}e^{16cT+5cT^{2}}+\int_{t}^{T}c^{\frac{1}{2}}\mathbb{E}\Big{[}\Big{|} u^{d,\varepsilon}(s,X_{s}^{d,t,x})-u^{d}(s,X_{s}^{d,t,x})\Big{|}\Big{]}\,ds\\ &\quad+T\varepsilon^{\frac{1}{2}}\left(3(cd^{c}T^{-1})^{\frac{1} {2}}\left(d^{c}+\|x\|^{2}\right)^{\frac{1}{2}}e^{4.5cT}\right)^{2}\\ &=8cd^{c}\varepsilon^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}} e^{16cT+5cT^{2}}+\int_{t}^{T}c^{\frac{1}{2}}\mathbb{E}\Big{[}\Big{|}u^{d, \varepsilon}(s,X_{s}^{d,t,x})-u^{d}(s,X_{s}^{d,t,x})\Big{|}\Big{]}\,ds\\ &\quad+9cd^{c}\varepsilon^{\frac{1}{2}}e^{9cT}(d^{c}+\|x\|^{2}) \\ &\leq 17cd^{c}\varepsilon^{\frac{1}{2}}(d^{c}+\|x\|^{2})e^{16cT+5 cT^{2}}+\int_{t}^{T}c\mathbb{E}\Big{[}\Big{|}u^{d,\varepsilon}(s,X_{s}^{d,t,x})-u^{d}(s,X_{s}^{d,t,x}) \Big{|}\Big{]}\,ds.\end{split} \tag{75}\]

This, (45), the disintegration theorem, and the flow property show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(s\in[0,T]\), \(t\in[s,T]\), \(x\in\mathbb{R}^{d}\) that

(76)
This, the triangle inequality, (30), (31), and (61) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(s\in[0,T]\), \(t\in[s,T]\), \(x\in\mathbb{R}^{d}\) that

\[\begin{split}&\mathbb{E}\Big{[}\Big{|}u^{d,\varepsilon}(t,X_{t}^{d,s,x})-u^{d}(t,X_{t}^{d,s,x})\Big{|}\Big{]}=\mathbb{E}\left[\Big{|}u^{d, \varepsilon}(t,\tilde{x})-u^{d}(t,\tilde{x})\Big{|}\Big{|}_{\tilde{x}=X_{t}^{ d,s,x}}\right]\\ &\leq\mathbb{E}\bigg{[}\mathbb{E}\Big{[}\Big{|}g_{\varepsilon}^{ d}(X_{T}^{d,\varepsilon,t,\tilde{x}})-g^{d}(X_{T}^{d,t,\tilde{x}})\Big{|} \Big{|}\Big{]}\Big{|}_{\tilde{x}=X_{t}^{d,s,x}}\bigg{]}\\ &\qquad+\int_{t}^{T}\mathbb{E}\left[\mathbb{E}\left[\Big{|}f_{ \varepsilon}(u^{d,\varepsilon}(r,X_{r}^{d,\varepsilon,t,\tilde{x}}))-f(u^{d}( r,X_{r}^{d,t,\tilde{x}}))\Big{|}\Big{]}\Big{|}\Big{|}_{\tilde{x}=X_{t}^{d,s,x}} \right]dr\\ &\leq 5(\varepsilon cd^{2c})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1 }{2}}e^{14cT+3cT^{2}}\\ &\qquad+17cd^{c}\varepsilon^{\frac{1}{2}}\left(d^{c}+\|x\|^{2} \right)e^{7cT}\right)e^{16cT+5cT^{2}}+c\int_{t}^{T}\mathbb{E}\left[\Big{|}u^{ d,\varepsilon}(r,X_{r}^{d,s,x})-u^{d}(r,X_{r}^{d,s,x})\Big{|}\Big{]}\,dr\\ &\leq 22cd^{c}\varepsilon^{\frac{1}{2}}(d^{c}+\|x\|^{2})e^{23cT+5cT^ {2}}+c\int_{t}^{T}\mathbb{E}\Big{[}\Big{|}u^{d,\varepsilon}(r,X_{r}^{d,s,x})- u^{d}(r,X_{r}^{d,s,x})\Big{|}\Big{]}\,dr.\end{split} \tag{77}\]

This, Gronwall's lemma, (29), and (45) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(s\in[0,T]\), \(t\in[s,T]\), \(x\in\mathbb{R}^{d}\) that

\[\begin{split}&\mathbb{E}\Big{[}\Big{|}u^{d,\varepsilon}(t,X_{t}^{d,s,x})-u^{d}(t,X_{t}^{d,s,x})\Big{|}\Big{]}\\ &\leq 22cd^{c}\varepsilon^{\frac{1}{2}}(d^{c}+\|x\|^{2})e^{23cT+5cT^{2} }\cdot e^{cT}=2cd^{c}\varepsilon^{\frac{1}{2}}(d^{c}+\|x\|^{2})e^{24cT+5cT^{2}} \end{split} \tag{78}\]

and hence \(\big{|}u^{d,\varepsilon}(t,x)-u^{d}(t,x)\big{|}\leq 2cd^{c}\varepsilon^{\frac{1}{2}} (d^{c}+\|x\|^{2})e^{24cT+5cT^{2}}\). This shows (iii).

The proof of Lemma 2.1 is thus completed. 

## 3. Euler-Maruyama and MLP approximations revisited

In Lemma 3.2 below we approximate the solution to the SFPE (86), associated to (83), through solution to the SFPE (85), associated to the Euler-Maruyama approximation (82).

**Setting 3.1**.: _Consider the notations given in Subsection 1.4, let \(T\in(0,\infty)\), \(c\in[1,\infty)\), for every \(K\in\mathbb{N}\) let \([\cdot]_{K}\colon\mathbb{R}\to\mathbb{R}\) satisfy for all \(t\in\mathbb{R}\) that \(\lfloor t\rfloor_{K}=\max(\{0,\frac{T}{K},\frac{2T}{K},\ldots,T\}\cap((-\infty, t)\cup\{0\}))\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), let \(\beta_{\varepsilon}^{d}\in C(\mathbb{R}^{d},\mathbb{R}^{d})\), \(\sigma_{\varepsilon}^{d}\in C(\mathbb{R}^{d},\mathbb{R}^{d\times d})\), \(f_{\varepsilon}\in C(\mathbb{R},\mathbb{R})\), \(g_{\varepsilon}^{d}\in C(\mathbb{R}^{d},\mathbb{R})\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(\gamma_{\varepsilon}^{d}\colon\mathbb{R}^{2d}\to\mathbb{R}^{d}\), be measurable, for every \(d\in\mathbb{N}\) let \(\nu^{d}\colon\mathcal{B}(\mathbb{R}^{d}\setminus\{0\})\to[0,\infty)\) be a Levy measure, and assume for all \(d\in\mathbb{N}\), \(x,y\in\mathbb{R}^{d}\), \(u_{1},w_{2}\in\mathbb{R}\), \(\varepsilon\in(0,1)\) that_

\[\Big{\|}\beta_{\varepsilon}^{d}(x)-\beta_{\varepsilon}^{d}(y)\Big{\|}^{2}+ \Big{\|}\sigma_{\varepsilon}^{d}(x)-\sigma_{\varepsilon}^{d}(y)\Big{\|}_{ \mathrm{F}}^{2}+\int_{\mathbb{R}^{d}\setminus\{0\}}\Big{\|}\gamma_{ \varepsilon}^{d}(x,z)-\gamma_{\varepsilon}^{d}(y,z)\Big{)}\Big{\|}^{2}\,\nu^{d }(dz)\leq c\|x-y\|^{2}, \tag{79}\]

\[|f_{\varepsilon}(w_{1})-f_{\varepsilon}(w_{2})|^{2}\leq c|w_{1}-w_{2}|^{2}, \quad\Big{|}g_{\varepsilon}^{d}(x)-g_{\varepsilon}^{d}(y)\Big{|}^{2}\leq cd^{c} T^{-1}\|x-y\|^{2}, \tag{80}\]

_and_

\[\Big{\|}\beta_{\varepsilon}^{d}(0)\Big{\|}^{2}+\Big{\|}\sigma_{\varepsilon}^{d }(0)\Big{\|}_{\mathrm{F}}^{2}+\int_{\mathbb{R}^{d}\setminus\{0\}}\Big{\|} \gamma_{\varepsilon}^{d}(0,z)\Big{\|}^{2}\,\nu^{d}(dz)+T^{3}|f_{\varepsilon}(0 )|^{2}+T|g_{\varepsilon}^{d}(0)|^{2}\leq cd^{c}. \tag{81}\]

**Lemma 3.2** (Discretization error).: _Assume Setting 3.1, for every \(d,K\in\mathbb{N}\), \(\theta\in\Theta\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\), \(t\in[0,T)\) let \((X_{s}^{d,K,\varepsilon,t,x})_{s\in[t,T]}\), \((X_{s}^{d,\varepsilon,t,x})_{s\in[t,T]}\) be adapted cadlag processes which satisfy for all \(s\in[t,T]\) that \(\mathbb{P}\)-a.s._

\[\begin{split} X_{s}^{d,K,\varepsilon,t,x}&=x+\int_{t}^{s} \beta_{\varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})\,dr+\int_{t}^ {s}\sigma_{\varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})\,dW_{r} ^{d}\\ &\qquad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\gamma_{ \varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})\,z)\,\tilde{N}^{d}( dr,dz),\end{split} \tag{82}\]

_and_

\[X_{s}^{d,\varepsilon,t,x}=x+\int_{t}^{s}\beta_{\varepsilon}^{d}(X_{r-}^{d, \varepsilon,t,x})dr+\int_{t}^{s}\sigma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t, x})dW_{r}^{d}+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\gamma_{\varepsilon}^{d}(X_{r-}^{d, \varepsilon,t,x},z)\tilde{N}^{d}(dr,dz), \tag{83}\]
_and for every \(d,K\in\mathbb{N}\), \(\varepsilon\in(0,1)\), let \(u^{d,K,\varepsilon},u^{d,\varepsilon}\colon[0,T]\times\mathds{R}^{d}\to\mathds{R}\) be measurable functions satisfying for all \(t\in[0,T]\), \(x\in\mathds{R}^{d}\) that_

\[\sup_{s\in[0,T]}\sup_{y\in\mathds{R}^{d}}\frac{|u^{d,K,\varepsilon}(s,y)|+|u^{ d,\varepsilon}(s,y)|}{1+\|y\|}<\infty, \tag{84}\]

\[u^{d,K,\varepsilon}(t,x)=\mathds{E}\Big{[}g_{\varepsilon}^{d}(X_{T}^{d,K, \varepsilon,t,x})\Big{]}+\int_{t}^{T}\mathds{E}\Big{[}f_{\varepsilon}(u^{d,K, \varepsilon}(r,X_{r}^{d,K,\varepsilon,t,x}))\Big{]}\,dr, \tag{85}\]

_and_

\[u^{d,\varepsilon}(t,x)=\mathds{E}\Big{[}g_{\varepsilon}^{d}(X_{T}^{d, \varepsilon,t,x})\Big{]}+\int_{t}^{T}\mathds{E}\Big{[}f_{\varepsilon}(u^{d, \varepsilon}(r,X_{r}^{d,\varepsilon,t,x}))\Big{]}\,dr. \tag{86}\]

_Then_

1. _for all_ \(d,K\in\mathbb{N}\)_,_ \(t\in[0,T]\)_,_ \(s\in[t,T]\)_,_ \(x\in\mathbb{R}^{d}\)_,_ \(\varepsilon\in(0,1)\) _we have that_ \[\mathbb{E}\bigg{[}d^{c}+\left\|X_{s}^{d,K,\varepsilon,t,x}\right\|^{2}\bigg{]} \leq(d^{c}+\|x\|^{2})e^{7c(s-t)},\] (87) _and_
2. _for all_ \(d,K\in\mathbb{N}\)_,_ \(t\in[0,T]\)_,_ \(x\in\mathds{R}^{d}\)_,_ \(\varepsilon\in(0,1)\)_, we have that_ \[\left|u^{d,K,\varepsilon}(t,x)-u^{d,\varepsilon}(t,x)\right|\leq 12c^{ \frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{21cT+5cT^{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2} }\frac{T^{\frac{1}{2}}}{K^{\frac{1}{2}}}.\] (88)

Proof of Lemma 3.2.: First, for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x,y\in\mathds{R}^{d}\) we have that

\[\mathbb{E}\bigg{[}d^{c}+\left\|X_{s}^{d,\varepsilon,t,x}\right\|^{2}\bigg{]} \leq(d^{c}+\|x\|^{2})e^{7c(s-t)} \tag{89}\]

and

\[\left|u^{d,\varepsilon}(t,x)-u^{d,\varepsilon}(t,y)\right|\leq 2(cd^{c}T^{-1})^{ \frac{1}{2}}\|x-y\|e^{5cT+2cT^{2}} \tag{90}\]

(cf. Lemma 2.1). Next, the triangle inequality, the fact that \(\forall\,a_{1},a_{2}\in\mathds{R}\colon(a_{1}+a_{2})^{2}\leq 2|a_{1}|^{2}+2|a_{2}|^ {2}\), (79), and (81) show for all \(d\in\mathbb{N}\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\|\beta_{\varepsilon}^{d}(x)\|^{2}\leq 2\|\beta_{\varepsilon}^{d}(0)\|^{2}+2\| \beta_{\varepsilon}^{d}(x)-\beta_{\varepsilon}^{d}(0)\|^{2}\leq 2cd^{c}+2c\|x\|^{2} =2c(d^{c}+\|x\|^{2}). \tag{91}\]

Similarly, we have for all \(d\in\mathbb{N}\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\left\|\sigma_{\varepsilon}^{d}(x)\right\|_{\mathrm{F}}^{2}\leq 2c(d^{c}+\|x\|^{2}). \tag{92}\]

Next, the triangle inequality, the fact that \(\forall\,a_{1},a_{2}\in\mathds{R}\colon(a_{1}+a_{2})^{2}\leq 2|a_{1}|^{2}+2|a_{2}|^ {2}\), (79), and (81) show for all \(d\in\mathbb{N}\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\int_{\mathds{R}^{d}\setminus\{0\}}\left\|\gamma_{ \varepsilon}^{d}(x,z)\right\|^{2}\nu^{d}(dz)&\leq\int_{\mathds{R }^{d}\setminus\{0\}}2\left\|\gamma_{\varepsilon}^{d}(0,z)\right\|^{2}+2\left\| \gamma_{\varepsilon}^{d}(x,z)-\gamma_{\varepsilon}^{d}(0,z)\right\|^{2}\nu^{d} (dz)\\ &\leq 2cd^{c}+2c\|x\|^{2}=2c(d^{c}+\|x\|^{2}).\end{split} \tag{93}\]

Next, Ito's formula (see, e.g., [20, Theorem 3.1]) and (82) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that \(\mathbb{P}\)-a.s.

\[\begin{split}\left\|X_{s}^{d,K,\varepsilon,t,x}\right\|^{2}& =\|x\|^{2}+\int_{t}^{s}\left(2\left\langle X_{\max\{t,\lfloor r \rfloor-_{K}\}}^{d,K,\varepsilon,t,x},\beta_{\varepsilon}^{d}(X_{\max\{t, \lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x})\right\rangle+\left\|\sigma_{ \varepsilon}^{d}(X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x}) \right\|_{\mathrm{F}}^{2}\right)dr\\ &\quad+2\int_{t}^{s}\sum_{i,j=1}^{d}\left(X_{\max\{t,\lfloor r \rfloor-_{K}\}}^{d,K,\varepsilon,t,x}\right)_{i}\left(\sigma_{\varepsilon}^{d} (X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x})\right)_{ij}d(W_{j}^ {d})_{r}\\ &\quad+2\int_{t}^{s}\int_{\mathds{R}^{d}\setminus\{0\}}\left\langle X _{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x},\gamma_{\varepsilon}^{d} (X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x},z)\right\rangle \tilde{N}^{d}(dz,dr)\\ &\quad+\int_{t}^{s}\int_{\mathds{R}^{d}\setminus\{0\}}\left\| \gamma_{\varepsilon}^{d}(X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x},z) \right\|^{2}N^{d}(dz,dr).\end{split} \tag{94}\]
Next, for every \(d,n,K\in\mathbb{N}\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) let \(\tau_{n}^{d,K,\varepsilon,x}\colon\Omega\to\mathbb{R}\) satisfy that

\[\begin{split}\tau_{n}^{d,K,\varepsilon,x}=\inf& \Bigg{\{}s\in[t,T]\colon\int_{t}^{s}\left(2\left\langle X_{\max\{t,[r-]_{K}\} ^{d,K,\varepsilon,t,x}}^{d,K,\varepsilon,t,x}\right\rangle\right)+\left\| \sigma_{\varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}^{d,K,\varepsilon,t,x}}^{d,K, \varepsilon,t,x})\right\|_{\mathrm{F}}^{2}\right)dr\\ &\qquad+\int_{t}^{s}\sum_{i,j=1}^{d}\left|\left(X_{\max\{t,[r-]_{ K}\}^{d,K,\varepsilon,t,x}}^{d,K,\varepsilon,t,x}\right)\right\rangle_{i} \left(\sigma_{\varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}^{d,K,\varepsilon,t,x}}^{d,K,\varepsilon,t,x})\right)\right|_{ij}^{2}dr\\ &\qquad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\sum_{i= 1}^{d}\left|\left(X_{\max\{t,[r-]_{K}\}^{d,K,\varepsilon,t,x}}^{d,K, \varepsilon,t,x}\right)\right\rangle_{i}\left(\gamma_{\varepsilon}^{d}(X_{ \max\{t,[r-]_{K}\}^{d,K,\varepsilon,t,x}}^{d,K,\varepsilon,t,x})\right) \right|_{i}^{2}\nu^{d}(dz)\,dr\\ &\qquad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\left\| \gamma_{\varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}^{d,K,\varepsilon,t,x}}^{d,K, \varepsilon,t,x})\right\|_{\mathrm{F}}^{2}\nu^{d}(dz)\,dr\geq n\Bigg{\}}\wedge T \end{split} \tag{95}\]

(with the convention that \(\inf\emptyset=\infty\)). Then (94), the fact that \(\forall\,d\in\mathbb{N},x,y\in\mathbb{R}^{d}\colon 2\langle x,y\rangle\leq\|x\|^{2}+\|y\|^{2}\), (91), (92), (93) show for all \(d,n,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

(96)

This, (35)-(37), and the fact that \(c\geq 1\) show for all \(d,K,n\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\max\biggl{\{}\mathbb{E}\biggl{[}d^{c}+\left\|X_{ \max\{t,[s\wedge\tau_{n}^{d,K,\varepsilon,t,x}}^{d,K,\varepsilon,t,x}\right\} ^{d,K,\varepsilon,t,x}\right\|^{2}\biggr{]}\biggr{]},\mathbb{E}\biggl{[}d^{c}+ \left\|X_{s\wedge\tau_{n}^{d,K,\varepsilon,t,x}}^{d,K,\varepsilon,t,x}\right\| ^{2}\biggr{]}\biggr{\}}\\ &\leq d^{c}+\|x\|^{2}+\mathbb{E}\Biggl{[}\int_{t}^{s\wedge\tau_{n} ^{d,K,\varepsilon,x}}\left(d^{c}+\left\|X_{\max\{t,[r]_{K}\}^{d,K,\varepsilon, t,x}}^{d,K,\varepsilon,t,x}\right\|^{2}\right)dr\Biggr{]}\\ &\qquad+\mathbb{E}\Biggl{[}\int_{t}^{s\wedge\tau_{n}^{d,K, \varepsilon,x}}6c\left(d^{c}+\left\|X_{\max\{t,[r]_{K}\}^{d,K,\varepsilon,t, x}}^{d,K,\varepsilon,t,x}\right\|^{2}\right)dr\Biggr{]}\\ &\leq d^{c}+\|x\|^{2}+7c\mathbb{E}\Biggl{[}\int_{t}^{s\wedge \tau_{n}^{d,K,\varepsilon,x}}\left(d^{c}+\left\|X_{\max\{t,[r]_{K}\}}^{d,K, \varepsilon,t,x}\right\|^{2}\right)dr\Biggr{]}\\ &\leq d^{c}+\|x\|^{2}+7c\int_{t}^{s}\mathbb{E}\biggl{[}d^{c}+ \left\|X_{\max\{t,[r\wedge\tau_{n}^{d,K,\varepsilon,t,x}}^{d,K,\varepsilon,t,x} \right\|_{K}}^{d,K,\varepsilon,t,x}\right\|^{2}\biggr{]}^{2}dr\biggr{]}\\ \end{split} \tag{97}\]

This, Fatou's lemma, and Gronwall's inequality show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\mathbb{E}\left[d^{c}+\left\|X_{\max\{t,[s]_{K}\} ^{d,K,\varepsilon,t,x}}^{d,K,\varepsilon,t,x}\right\|^{2}\right]& \leq\liminf_{n\to\infty}\mathbb{E}\biggl{[}d^{c}+\left\|X_{\max\{t,[s \wedge\tau_{n}^{d,K,\varepsilon,t,x}}^{d,K,\varepsilon,t,x}\right\|_{K}}^{d,K, \varepsilon,t,x}\right\|^{2}\biggr{]}\\ &\leq(d^{c}+\|x\|^{2})e^{7\varepsilon(s-t)}.\end{split} \tag{98}\]
This, Fatou's lemma, (97), and the fact that \(\forall\,t\in[0,T],s\in[t,T]\colon 1+7c\int_{t}^{s}e^{7c(r-t)}\,dr=1+e^{7c(r-t)} |_{r=s}^{t}=e^{7c(s-t)}\) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\mathbb{E}\left[d^{c}+\left\|X_{s}^{d,K, \varepsilon,t,x}\right\|^{2}\right]&\leq\liminf_{n\to\infty} \mathbb{E}\left[d^{c}+\left\|X_{s\wedge\tau_{n}^{d,K,\varepsilon,t,x}}^{d,K, \varepsilon,t,x}\right\|^{2}\right]\\ &\leq d^{c}+\|x\|^{2}+7c\mathbb{E}\left[\int_{t}^{s}\left(d^{c}+ \left\|X_{\max\{t,\lceil r\rfloor_{K}\}}^{d,K,\varepsilon,t,x}\right\|^{2} \right)dr\right]\\ &\leq d^{c}+\|x\|^{2}+7c\int_{t}^{s}(d^{c}+\|x\|^{2})e^{7c(r-t)} \,dr\\ &=(d^{c}+\|x\|^{2})(1+7c\int_{t}^{s}e^{7c(r-t)}\,dr)\\ &=(d^{c}+\|x\|^{2})e^{7c(s-t)}.\end{split} \tag{99}\]

This shows (i).

Next, Holder's inequality, (91), and (99) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s,s^{\prime}\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\mathbb{E}\left[\left\|\int_{s}^{s^{\prime}}\beta_{ \varepsilon}^{d}(X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x}) \,dr\right\|^{2}\right]&\leq\mathbb{E}\left[\left(\int_{s}^{s^{ \prime}}\left\|\beta_{\varepsilon}^{d}(X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x})\right\|dr\right)^{2}\right]\\ &\leq\mathbb{E}\left[|s^{\prime}-s|\int_{s}^{s^{\prime}}\left\| \beta_{\varepsilon}^{d}(X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon, t,x})\right\|^{2}dr\right]\\ &\leq T|s^{\prime}-s|\sup_{r\in[s,s^{\prime}]}\mathbb{E}\left[2c \left(d^{c}+\left\|X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x} \right\|^{2}\right)\right]\\ &\leq T|s^{\prime}-s|\cdot 2c(d^{c}+\|x\|^{2})e^{7cT}.\end{split} \tag{100}\]

Next, Ito's isometry, (92), and (99) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s,s^{\prime}\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\mathbb{E}\left[\left\|\int_{s}^{s^{\prime}}\sigma_{ \varepsilon}^{d}(X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x}) \,dW_{r}^{d}\right\|^{2}\right]&=\mathbb{E}\left[\int_{s}^{s^{ \prime}}\left\|\sigma_{\varepsilon}^{d}(X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{ d,K,\varepsilon,t,x})\right\|_{\mathbb{F}}^{2}dr\right]\\ &\leq|s^{\prime}-s|\sup_{r\in[s,s^{\prime}]}\mathbb{E}\left[2c \left(d^{c}+\left\|X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x} \right\|^{2}\right)\right]\\ &\leq|s^{\prime}-s|\cdot 2c(d^{c}+\|x\|^{2})e^{7cT}.\end{split} \tag{101}\]

Next, Ito's isometry, (93), and (99) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s,s^{\prime}\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\mathbb{E}\left[\left\|\int_{s}^{s^{\prime}}\int_{ \mathbb{R}^{d}\setminus\{0\}}^{d}\gamma_{\varepsilon}^{d}(X_{\max\{t,\lfloor r -\rfloor_{K}\}}^{d,K,\varepsilon,t,x},z)\,\tilde{N}^{d}(dr,dz)\right\|^{2} \right]\\ &=\mathbb{E}\left[\int_{s}^{s^{\prime}}\int_{\mathbb{R}^{d} \setminus\{0\}}\left\|\gamma_{\varepsilon}^{d}(X_{\max\{t,\lfloor r-\rfloor_{ K}\}}^{d,K,\varepsilon,t,x},z)\right\|^{2}\nu^{d}(dz)\,dr\right]\\ &\leq|s^{\prime}-s|\sup_{r\in[s,s^{\prime}]}\mathbb{E}\left[2c \left(d^{c}+\left\|X_{\max\{t,\lfloor r-\rfloor_{K}\}}^{d,K,\varepsilon,t,x} \right\|^{2}\right)\right]\\ &\leq|s^{\prime}-s|\cdot 2c(d^{c}+\|x\|^{2})e^{7cT}.\end{split} \tag{102}\]
This, (82), the fact that \(\forall\,d\in\mathbb{N},x,y,z\in\mathds{R}^{d}\colon\|x+y+z\|^{2}\leq 3\|x\|^{2}+3 \|y\|^{2}+3\|z\|^{2}\), (100), and (101) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s,s^{\prime}\in[t,T]\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\mathbb{E}\bigg{[}\Big{\|}X_{s^{\prime}}^{d,K,\varepsilon,t,x}-X _{s}^{d,K,\varepsilon,t,x}\Big{\|}^{2}\bigg{]} \tag{103}\] \[\leq 3\mathbb{E}\left[\left\|\int_{s}^{s^{\prime}}\beta_{ \varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})\,dr\right\|^{ 2}\right]+3\mathbb{E}\left[\left\|\int_{s}^{s^{\prime}}\sigma_{\varepsilon}^{d }(X_{\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})\,dW_{r}^{d}\right\|^{2}\right]\] \[\quad+3\mathbb{E}\left[\left\|\int_{s}^{s^{\prime}}\int_{\mathbb{ R}^{d}\setminus\{0\}}\gamma_{\varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}}^{d,K, \varepsilon,t,x},z)\,\tilde{N}^{d}(dr,dz)\right\|^{2}\right]\] \[\leq 3\cdot T|s^{\prime}-s|\cdot 2c(d^{c}+\|x\|^{2})e^{7cT}+3 \cdot|s^{\prime}-s|\cdot 2c(d^{c}+\|x\|^{2})e^{7cT}\] \[\quad+3\cdot|s^{\prime}-s|\cdot 2c(d^{c}+\|x\|^{2})e^{7cT}\] \[=6c(T+2)e^{7cT}(d^{c}+\|x\|^{2})|s^{\prime}-s|.\]

This, Holder's inequality, (79), the fact that \(\forall\,d\in\mathbb{N},x,y\in\mathds{R}^{d}\colon\|x+y\|^{2}\leq 2\|x\|^{2}+2\|y\|^ {2}\), and (103) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\mathbb{E}\left[\left\|\int_{t}^{s}\left(\beta_{\varepsilon}^{d }(X_{\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})-\beta_{\varepsilon}^{d}(X_{r -}^{d,\varepsilon,t,x})\right)dr\right\|^{2}\right] \tag{104}\] \[\leq\mathbb{E}\bigg{[}\left(\int_{t}^{s}\left\|\beta_{\varepsilon }^{d}(X_{\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})-\beta_{\varepsilon}^{d}(X_ {r-}^{d,\varepsilon,t,x})\right\|dr\right)^{2}\bigg{]}\] \[\leq T\mathbb{E}\bigg{[}\int_{t}^{s}\left\|\beta_{\varepsilon}^{ d}(X_{\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})-\beta_{\varepsilon}^{d}(X_{r -}^{d,\varepsilon,t,x})\right\|^{2}dr\bigg{]}\] \[\leq T\mathbb{E}\bigg{[}\int_{t}^{s}c\left\|X_{\max\{t,[r-]_{K} \}}^{d,K,\varepsilon,t,x}-X_{r-}^{d,\varepsilon,t,x}\right\|^{2}dr\bigg{]}\] \[\leq 2cT\int_{t}^{s}\mathbb{E}\left[\left\|X_{\max\{t,[r]_{K}\}} ^{d,K,\varepsilon,t,x}-X_{r}^{d,\varepsilon,t,x}\right\|^{2}\right]+2cT\int_{t }^{s}\mathbb{E}\left[\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d,\varepsilon, t,x}\right\|^{2}\right]dr\] \[\leq 2cT(s-t)\cdot 6c(T+2)e^{7cT}(d^{c}+\|x\|^{2})\frac{T}{K}+2cT \int_{t}^{s}\mathbb{E}\bigg{[}\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d, \varepsilon,t,x}\right\|^{2}\bigg{]}\,dr\] \[\leq 12c^{2}T^{2}(T+2)e^{7cT}(d^{c}+\|x\|^{2})\frac{T}{K}+2cT\int_{t }^{s}\mathbb{E}\bigg{[}\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d,\varepsilon,t,x}\right\|^{2}\bigg{]}\,dr\]

Next, Ito's isometry, (79), the fact that \(\forall\,d\in\mathbb{N},x,y\in\mathds{R}^{d}\colon\|x+y\|^{2}\leq 2\|x\|^{2}+2\|y\|^ {2}\), and (103) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\mathbb{E}\left[\left\|\int_{t}^{s}\left(\sigma_{\varepsilon}^{d }(X_{\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})-\sigma_{\varepsilon}^{d}(X_{r -}^{d,\varepsilon,t,x})\right)dW_{r}^{d}\right\|^{2}\right] \tag{105}\] \[=\mathbb{E}\bigg{[}\int_{t}^{s}\left\|\sigma_{\varepsilon}^{d}(X_ {\max\{t,[r-]_{K}\}}^{d,K,\varepsilon,t,x})-\sigma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x})\right\|^{2}dr\bigg{]}\] \[\leq\mathbb{E}\bigg{[}\int_{t}^{s}c\left\|X_{\max\{t,[r-]_{K}\}} ^{d,K,\varepsilon,t,x}-X_{r-}^{d,\varepsilon,t,x}\right\|^{2}dr\bigg{]}\] \[\leq 2c\int_{t}^{s}\mathbb{E}\bigg{[}\left\|X_{\max\{t,[r]_{K}\}} ^{d,K,\varepsilon,t,x}-X_{r}^{d,K,\varepsilon,t,x}\right\|^{2}\bigg{]}\,dr+2c \int_{t}^{s}\mathbb{E}\bigg{[}\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d, \varepsilon,t,x}\right\|^{2}\bigg{]}\,dr\] \[\leq 2c(s-t)\cdot 6c(T+2)e^{7cT}(d^{c}+\|x\|^{2})\frac{T}{K}+2c\int_{t }^{s}\mathbb{E}\bigg{[}\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d,\varepsilon, t,x}\right\|^{2}\bigg{]}\,dr\] \[=12c^{2}T(T+2)e^{7cT}(d^{c}+\|x\|^{2})\frac{T}{K}+2c\int_{t}^{s} \mathbb{E}\bigg{[}\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d,\varepsilon,t,x} \right\|^{2}\bigg{]}\,dr.\]
Next, Ito's isometry, (79), the fact that \(\forall\,d\in\mathbb{N},x,y,z\in\mathds{R}^{d}\colon\|x+y\|^{2}\leq 2\|x\|^{2}+ 2\|y\|^{2}\), and (103) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\left[\left\|\int_{t}^{s}\int_{\mathds{R} ^{d}\setminus\{0\}}\left(\gamma_{\varepsilon}^{d}(X_{\max\{t,|r-|_{K}\}}^{d,K,\varepsilon,t,x},z)-\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z) \right)\tilde{N}^{d}(dr,dz)\right\|^{2}\right]\\ &=\mathbb{E}\left[\int_{t}^{s}\int_{\mathds{R}^{d}\setminus\{0\}} \left\|\gamma_{\varepsilon}^{d}(X_{\max\{t,|r-|_{K}\}}^{d,K,\varepsilon,t,x},z )-\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)\right\|^{2}\nu^{d}( dz)dr\right]\\ &\leq\mathbb{E}\left[\int_{t}^{s}c\left\|X_{\max\{t,|r-|_{K}\}}^ {d,K,\varepsilon,t,x}-X_{r-}^{d,\varepsilon,t,x}\right\|^{2}dr\right]\\ &\leq 2c\int_{t}^{s}\mathbb{E}\left[\left\|X_{\max\{t,|r\}}^{d,K, \varepsilon,t,x}-X_{r}^{d,K,\varepsilon,t,x}\right\|^{2}\right]dr+2c\int_{t}^ {s}\mathbb{E}\left[\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d,\varepsilon,t,x}\right\|^{2}\right]dr\\ &\leq 2c(s-t)\cdot 6c(T+2)e^{\tau cT}(d^{c}+\|x\|^{2})\frac{T}{K }+2c\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d, \varepsilon,t,x}\right\|^{2}\right]dr\\ &\leq 12c^{2}T(T+2)e^{\tau cT}(d^{c}+\|x\|^{2})\frac{T}{K}+2c\int_{ t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d,\varepsilon,t,x}\right\|^{2}\right]dr.\end{split} \tag{106}\]

This, the fact that \(\forall\,d\in\mathbb{N},x,y,z\in\mathds{R}^{d}\colon\|x+y+z\|^{2}\leq 3\|x\|^{2}+ 3\|y\|^{2}+3\|z\|^{2}\), (104), (105) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\left[\left\|X_{s}^{d,K,\varepsilon,t,x}-X_{ s}^{d,\varepsilon,t,x}\right\|^{2}\right]\\ &\leq 3\mathbb{E}\left[\left\|\int_{t}^{s}\left(\beta_{ \varepsilon}^{d}(X_{\max\{t,|r-|_{K}\}}^{d,K,\varepsilon,t,x})-\beta_{ \varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x})\right)dr\right\|^{2}\right]\\ &\quad+3\mathbb{E}\left[\left\|\int_{t}^{s}\left(\sigma_{ \varepsilon}^{d}(X_{\max\{t,|r-|_{K}\}}^{d,K,\varepsilon,t,x})-\sigma_{ \varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x})\right)dW_{r}^{d}\right\|^{2} \right]\\ &\quad+3\mathbb{E}\left[\left\|\int_{t}^{s}\int_{\mathds{R}^{d} \setminus\{0\}}\left(\gamma_{\varepsilon}^{d}(X_{\max\{t,|r-|_{K}\}}^{d,K, \varepsilon,t,x},z)-\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z) \right)\tilde{N}^{d}(dr,dz)\right\|^{2}\right]\end{split} \tag{107}\]

and

\[\begin{split}&\mathbb{E}\bigg{[}\left\|X_{s}^{d,K, \varepsilon,t,x}-X_{s}^{d,\varepsilon,t,x}\right\|^{2}\bigg{]}\\ &\leq 3\left[12c^{2}T^{2}(T+2)e^{\tau cT}(d^{c}+\|x\|^{2})\frac{T}{K }+2cT\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d, \varepsilon,t,x}\right\|^{2}\right]dr\right]\\ &\quad+3\left[12c^{2}T(T+2)e^{\tau cT}(d^{c}+\|x\|^{2})\frac{T}{K }+2c\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d, \varepsilon,t,x}\right\|^{2}\right]dr\right]\\ &\quad+3\left[12c^{2}T(T+2)e^{\tau cT}(d^{c}+\|x\|^{2})\frac{T}{K }+2c\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d, \varepsilon,t,x}\right\|^{2}\right]dr\right]\\ &=36c^{2}T(T+2)^{2}e^{\tau cT}(d^{c}+\|x\|^{2})\frac{T}{K}+6c(T+2 )\int_{t}^{s}\mathbb{E}\left[\left\|X_{r}^{d,K,\varepsilon,t,x}-X_{r}^{d, \varepsilon,t,x}\right\|^{2}\right]dr.\end{split} \tag{108}\]

This, (89), (99), and Gronwall's inequality show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\mathbb{E}\left[\left\|X_{s}^{d,K,\varepsilon,t,x}-X_{s}^{d,\varepsilon,t,x} \right\|^{2}\right]\leq 36c^{2}T(T+2)^{2}e^{\tau cT}(d^{c}+\|x\|^{2})\frac{T}{K}e^{6c(T+2)T}. \tag{109}\]
This, (80), and Jensen's inequality show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\mathbb{E}\Big{[}\Big{|}g_{\varepsilon}^{d}(X_{T}^{d,K,\varepsilon,t,x})-g_{\varepsilon}^{d}(X_{T}^{d,\varepsilon,t,x})\Big{|} \Big{]}&\leq(cd^{c})^{\frac{1}{2}}T^{-\frac{1}{2}}\mathbb{E} \Big{[}\Big{\|}X_{T}^{d,K,\varepsilon,t,x}-X_{T}^{d,\varepsilon,t,x}\Big{\|} \Big{]}\\ &\leq(cd^{c})^{\frac{1}{2}}T^{-\frac{1}{2}}\left(\mathbb{E} \left[\left\|X_{T}^{d,K,\varepsilon,t,x}-X_{T}^{d,\varepsilon,t,x}\right\|^{2 }\right]\right)^{\frac{1}{2}}\\ &\leq(cd^{c})^{\frac{1}{2}}T^{-\frac{1}{2}}\left(36c^{2}T(T+2)^{ 2}e^{7cT}(d^{c}+\|x\|^{2})\frac{T}{K}e^{6c(T+2)T}\right)^{\frac{1}{2}}\\ &\leq 6c^{\frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{10cT+3cT^{2}}(d^{c}+\|x \|^{2})^{\frac{1}{2}}\frac{T^{\frac{1}{2}}}{K^{\frac{1}{2}}}.\end{split} \tag{110}\]

Next, (90), Jensen's inequality, and (109) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(r\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\Big{[}\Big{|}u^{d,\varepsilon}(r,X_{r}^{d,K,\varepsilon,t,x})-u^{d,\varepsilon}(r,X_{r}^{d,\varepsilon,t,x})\Big{|} \Big{]}\\ &\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\mathbb{E}\Big{[}\Big{\|}X_{r}^{ d,K,\varepsilon,t,x}-X_{r}^{d,\varepsilon,t,x}\Big{\|}\Big{]}\,e^{5cT+2cT^{2}}\\ &\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\left(\mathbb{E}\left[\left\|X_{ r}^{d,K,\varepsilon,t,x}-X_{r}^{d,\varepsilon,t,x}\right\|^{2}\right]\right)^{ \frac{1}{2}}e^{5cT+2cT^{2}}\\ &\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\left(36c^{2}T(T+2)^{2}e^{7cT}(d ^{c}+\|x\|^{2})\frac{T}{K}e^{6c(T+2)T}\right)^{\frac{1}{2}}e^{5cT+2cT^{2}}\\ &\leq 12c^{\frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{15cT+5cT^{2}}(d^{c}+\|x \|^{2})^{\frac{1}{2}}\frac{T^{\frac{1}{2}}}{K^{\frac{1}{2}}}.\end{split} \tag{111}\]

Next, Jensen's inequality and (99) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(r\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\mathbb{E}\Big{[}\Big{|}u^{d,K,\varepsilon}(r,X_{r}^{ d,K,\varepsilon,t,x})-u^{d,\varepsilon}(r,X_{r}^{d,K,\varepsilon,t,x})\Big{|} \Big{]}\\ &\leq e^{3.5c(T-r)}\mathbb{E}\Bigg{[}\bigg{(}d^{c}+\Big{\|}X_{r} ^{d,K,\varepsilon,t,x}\Big{\|}^{2}\bigg{)}^{\frac{1}{2}}\sup_{y\in\mathbb{R}^{ d}}\,\frac{\big{|}u^{d,K,\varepsilon}(r,y)-u^{d,\varepsilon}(r,y)\big{|}}{e^{3.5c (T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}\\ &\leq e^{3.5c(T-r)}\left(\mathbb{E}\left[d^{c}+\Big{\|}X_{r}^{ d,K,\varepsilon,t,x}\Big{\|}^{2}\right]\right)^{\frac{1}{2}}\sup_{y\in\mathbb{R}^{ d}}\,\frac{\big{|}u^{d,K,\varepsilon}(r,y)-u^{d,\varepsilon}(r,y)\big{|}}{e^{3.5c (T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}\\ &\leq e^{3.5c(T-r)}(d^{c}+\|x\|^{2})^{\frac{1}{2}}e^{3.5c(r-t)} \sup_{y\in\mathbb{R}^{d}}\,\frac{\big{|}u^{d,K,\varepsilon}(r,y)-u^{d, \varepsilon}(r,y)\big{|}}{e^{3.5c(T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}\\ &=(d^{c}+\|x\|^{2})^{\frac{1}{2}}e^{3.5c(T-t)}\sup_{y\in\mathbb{R }^{d}}\,\frac{\big{|}u^{d,K,\varepsilon}(r,y)-u^{d,\varepsilon}(r,y)\big{|}}{e^{ 3.5c(T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}.\end{split} \tag{112}\]

This, the triangle inequality, (80), the fact that \(c\geq 1\), (110), (111), the fact that \(1+cT\leq e^{cT}\) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\Big{|}u^{d,K,\varepsilon}(t,x)-u^{d,\varepsilon}(t,x) \Big{|}\\ &\leq\mathbb{E}\Big{[}\Big{|}g_{\varepsilon}^{d}(X_{T}^{d,K, \varepsilon,t,x})-g_{\varepsilon}^{d}(X_{T}^{d,\varepsilon,t,x})\Big{|} \Big{]}+\int_{t}^{T}\mathbb{E}\Big{[}\Big{|}f_{\varepsilon}(u^{d,K, \varepsilon}(r,X_{r}^{d,K,\varepsilon,t,x}))-f_{\varepsilon}(u^{d,\varepsilon} (r,X_{r}^{d,\varepsilon,t,x}))\Big{|}\Big{]}\,dr\\ &\leq\mathbb{E}\Big{[}\Big{|}g_{\varepsilon}^{d}(X_{T}^{d,K, \varepsilon,t,x})-g_{\varepsilon}^{d}(X_{T}^{d,\varepsilon,t,x})\Big{|}\Big{]}+ \int_{t}^{T}c\mathbb{E}\Big{[}\Big{|}u^{d,K,\varepsilon}(r,X_{r}^{d,K, \varepsilon,t,x})-u^{d,\varepsilon}(r,X_{r}^{d,K,\varepsilon,t,x})\Big{|}\Big{]} \\ &\quad+\int_{t}^{T}c\mathbb{E}\Big{[}\Big{|}u^{d,\varepsilon}(r,X_ {r}^{d,K,\varepsilon,t,x})-u^{d,\varepsilon}(r,X_{r}^{d,\varepsilon,t,x})\Big{|} \Big{]}\,dr\end{split} \tag{113}\]
and

\[\begin{split}\left|u^{d,K,\varepsilon}(t,x)-u^{d,\varepsilon}(t,x) \right|&\leq 6c^{\frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{10cT+3cT^{2}}(d^{c}+\|x\|^{2})^{ \frac{1}{2}}\frac{T^{\frac{1}{2}}}{K^{\frac{1}{2}}}\\ &\quad+\int_{t}^{T}c(d^{c}+\|x\|^{2})^{\frac{1}{2}}e^{3.5c(T-t)} \sup_{y\in\mathbb{R}^{d}}\frac{\left|u^{d,K,\varepsilon}(r,y)-u^{d, \varepsilon}(r,y)\right|}{e^{3.5c(T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}dr\\ &\quad+Tc\cdot 12c^{\frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{15cT+5cT^{2}} (d^{c}+\|x\|^{2})^{\frac{1}{2}}\frac{T^{\frac{1}{2}}}{K^{\frac{1}{2}}}\\ &\leq 12c^{\frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{16cT+5cT^{2}}(d^{c}+\|x \|^{2})^{\frac{1}{2}}\frac{T^{\frac{1}{2}}}{K^{\frac{1}{2}}}\\ &\quad+\int_{t}^{T}c(d^{c}+\|x\|^{2})^{\frac{1}{2}}e^{3.5c(T-t)} \sup_{y\in\mathbb{R}^{d}}\frac{\left|u^{d,K,\varepsilon}(r,y)-u^{d, \varepsilon}(r,y)\right|}{e^{3.5c(T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}\,dr. \end{split} \tag{114}\]

Dividing by \((d^{c}+\|x\|^{2})^{\frac{1}{2}}e^{3.5c(T-t)}\) shows for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\sup_{x\in\mathbb{R}^{d}}\frac{\left|u^{d,K, \varepsilon}(t,x)-u^{d,\varepsilon}(t,x)\right|}{e^{3.5c(T-t)}(d^{c}+\|y\|^{2 })^{\frac{1}{2}}}\\ \leq 12c^{\frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{16cT+5cT^{2}} \frac{T^{\frac{1}{2}}}{K^{\frac{1}{2}}}+\int_{t}^{T}c\sup_{y\in\mathbb{R}^{d} }\frac{\left|u^{d,K,\varepsilon}(r,y)-u^{d,\varepsilon}(r,y)\right|}{e^{3.5c( T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}\,dr.\end{split} \tag{115}\]

This, (84), (89), (99), and Gronwall's inequality show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(\varepsilon\in(0,1)\) that

\[\begin{split}\sup_{y\in\mathbb{R}^{d}}\frac{\left|u^{d,K, \varepsilon}(t,y)-u^{d,\varepsilon}(t,y)\right|}{e^{3.5c(T-t)}(d^{c}+\|y\|^{2 })^{\frac{1}{2}}}&\leq 12c^{\frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{16cT+5cT^{2}} \frac{T^{\frac{1}{2}}}{K^{\frac{1}{2}}}\cdot e^{cT}\\ &=12c^{\frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{17cT+5cT^{2}}\frac{T^{ \frac{1}{2}}}{K^{\frac{1}{2}}}.\end{split} \tag{116}\]

Hence, for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) we have that

\[\left|u^{d,K,\varepsilon}(t,x)-u^{d,\varepsilon}(t,x)\right|\leq 12c^{\frac{3}{2}} d^{\frac{5}{2}}(T+2)e^{21cT+5cT^{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}\frac{T^{ \frac{1}{2}}}{K^{\frac{1}{2}}}. \tag{117}\]

This shows (ii). The proof of Lemma 3.2 is thus completed. 

In Lemma 3.3 below we approximate the solution to SFPE (120), associated to (118), by the MLP approximation (121).

**Lemma 3.3**.: _Assume Setting 3.1, let \((\Omega,\mathcal{F},\mathbb{P}\,,(\mathbb{F}_{t})_{t\in[0,T]})\) be a probability space satisfying the usual conditions, let \(\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}\), for every \(d\in\mathbb{N}\) let \(W^{d,\theta}\colon\Omega\times[0,T]\to\mathbb{R}^{d}\), \(\theta\in\Theta\), be identically independently distributed standard \((\mathbb{F}_{t})_{t\in[0,T]}\)-Brownian motions, for every \(d\in\mathbb{N}\) let \(N^{d,\theta}\), \(\theta\in\Theta\), be independent \((\mathbb{F}_{t})_{t\in[0,T]}\)-Poisson random measures on \([0,\infty)\times(\mathbb{R}^{d}\setminus\{0\})\) with intensity \(\nu^{d}\), for every \(d\in\mathbb{N}\), \(\theta\in\Theta\) let \(\tilde{N}^{d,\theta}(dt,dz)=N^{d,\theta}(dt,dz)-dt\,\nu^{d}(dz)\), assume for all \(d\in\mathbb{N}\) that \(\mathcal{F}_{0}\), \((N^{d,\theta})_{\theta\in\Theta}\) and \((W^{d,\theta})_{\theta\in\Theta}\), are independent, for every \(d,K\in\mathbb{N}\), \(\theta\in\Theta\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\), \(t\in[0,T)\) let \((X_{s}^{d,\theta,K,\varepsilon,t,x})_{s\in[t,T]}\) satisfy that \(X_{t}^{d,\theta,K,\varepsilon,t,x}=x\) and_

\[\begin{split} X_{s}^{d,\theta,K,\varepsilon,t,x}&=x+ \int_{t}^{s}\beta_{\varepsilon}^{d}(X_{\max\{t,[r^{-}]_{K}\}}^{d,\theta,K, \varepsilon,t,x})\,dr+\int_{t}^{s}\sigma_{\varepsilon}^{d}(X_{\max\{t,[r^{-}]_{ K}\}}^{d,\theta,K,\varepsilon,t,x})\,dW_{r}^{d,\theta}\\ &\quad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\sigma_{ \varepsilon}^{d}(X_{\max\{t,[r^{-}]_{K}\}}^{d,\theta,K,\varepsilon,t,x},z)\, \tilde{N}^{d,\theta}(dr,dz),\end{split} \tag{118}\]

_for every \(d,K\in\mathbb{N}\), \(\varepsilon\in(0,1)\), let \(u^{d,K,\varepsilon}\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}\) be measurable functions satisfying for all \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) that \(\mathbb{E}\left[\left|g_{\varepsilon}^{d}(X_{t,T}^{d,0,K,\varepsilon,x})\right| \right]+\int_{t}^{T}\mathbb{E}\Big{[}\left|f_{\varepsilon}(u^{d,K,\varepsilon} (r,X_{t,r}^{d,0,K,\varepsilon,x}))\right|\Big{]}<\infty,\)_

\[\sup_{s\in[0,T]}\sup_{y\in\mathbb{R}^{d}}\frac{\left|u^{d,K,\varepsilon}(s,y) \right|}{1+\|y\|}<\infty \tag{119}\]
_and_

\[u^{d,K,\varepsilon}(t,x)=\mathbb{E}\left[g_{\varepsilon}^{d}(X_{T}^{d,0,K, \varepsilon,t,x})\right]+\int_{t}^{T}\mathbb{E}\Big{[}f_{\varepsilon}(u^{d,K, \varepsilon}(r,X_{r}^{d,0,K,\varepsilon,t,x}))\Big{]}\,dr, \tag{120}\]

_let \(\theta:\Omega\to[0,1]\), \(\theta\in\Theta\), be i.i.d random variables which satisfy for all \(t\in(0,1)\) that \(\mathbb{P}\left(\mathfrak{t}^{0}\leq t\right)=t\), for every \(\theta\in\Theta\), \(t\in[0,T]\) let \(\mathfrak{T}_{t}^{\theta}:\Omega\to\mathbb{R}\) satisfy for all \(\theta\in\Theta\) that \(\mathfrak{T}_{t}^{\theta}=t+(T-t)\mathfrak{t}^{\theta}\), assume for all \(d\in\mathbb{N}\) that \((\mathfrak{t}^{\theta})_{\theta\in\Theta}\), \((N^{d,\theta})_{\theta\in\Theta}\) and \((W^{d,\theta})_{\theta\in\Theta}\), are independent, for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(U_{n,m}^{d,\theta,K,\varepsilon}\colon[0,T]\times\mathbb{R}^{d}\times\Omega \to\mathbb{R}\), \(\theta\in\Theta\), \(n,m\in\mathbb{Z}\), satisfy for all \(\theta\in\Theta\), \(n\in\mathbb{N}_{0}\), \(m\in\mathbb{N}\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) that_

\[U_{n,m}^{d,\theta,K,\varepsilon}(t,x)=\frac{\mathbbm{1}_{\mathbb{ N}}(n)}{m^{n}}\sum_{i=1}^{m^{n}}g_{\varepsilon}^{d}\Big{(}X_{T}^{d,(\theta,0,-i),K, \varepsilon,t,x}\Big{)} \tag{121}\] \[\quad+\sum_{\ell=0}^{n-1}\frac{(T-t)}{m^{n-\ell}}\sum_{i=1}^{m^{ n-\ell}}\Big{(}f_{\varepsilon}\circ U_{\ell,m}^{d,(\theta,\ell,i),K,\varepsilon}- \mathbbm{1}_{\mathbb{N}}(\ell)f_{\varepsilon}\circ U_{\ell-1,m}^{d,(\theta,- \ell,i),K,\varepsilon}\Big{)}\Big{(}\mathfrak{T}_{t}^{(\theta,\ell,i)},X_{ \mathfrak{T}_{t}^{(\theta,\ell,i)}}^{d,(\theta,\ell,i),K,\varepsilon,t,x} \Big{)}.\]

_Then for all \(d,K,n,m\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) we have that \(U_{n,m}^{d,\theta,K,\varepsilon}\) is measurable and \(\mathbb{E}\bigg{[}\bigg{(}\Big{|}U_{n,m}^{d,\theta,K,\varepsilon}(t,x)-u^{d, K,\varepsilon}(t,x)\Big{|}^{2}\bigg{]}\bigg{)}^{\frac{1}{2}}\leq 6e^{\frac{m}{2}}m^{-\frac{n}{2}}e^{12cTn} (cd^{c}T^{-1})^{\frac{1}{2}}\left(d^{c}+\|x\|^{2}\right)^{\frac{1}{2}}.\)_

Proof of Lemma 3.3.: For measurability see [23, Lemma 3.2]. Next, (25) and the triangle inequality show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(x\in\mathbb{R}^{d}\), \(w\in\mathbb{R}\) that

\[\Big{|}g_{\varepsilon}^{d}(x)\Big{|}\leq\Big{|}g_{\varepsilon}^{d}(0)\Big{|}+( cd^{c}T^{-1})^{\frac{1}{2}}\|x\|\leq(cd^{c}T^{-1})^{\frac{1}{2}}+(cd^{c}T^{-1})^{ \frac{1}{2}}\|x\|\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}} \tag{122}\]

and

\[|f_{\varepsilon}(w)|\leq|f_{\varepsilon}(0)|+c^{\frac{1}{2}}|w|\leq(cd^{c}T^{ -3})^{\frac{1}{2}}+c^{\frac{1}{2}}|w|. \tag{123}\]

First, for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(s\in[t,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) we have that

\[\mathbb{E}\bigg{[}d^{c}+\left\|X_{t,s}^{d,0,K,\varepsilon,x}\right\|^{2} \bigg{]}\leq(d^{c}+\|x\|^{2})e^{7c(s-t)} \tag{124}\]

(cf. Lemma 3.2). This, (122), and Jensen's inequality show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\)\(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\mathbb{E}\Big{[}g_{\varepsilon}^{d}(X_{T}^{d,0,K,\varepsilon,t,x}) \Big{]} \leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\mathbb{E}\Bigg{[}\bigg{(}d^{c}+ \left\|X_{T}^{d,0,K,\varepsilon,t,x}\right\|^{2}\bigg{)}^{\frac{1}{2}}\Bigg{]} \tag{125}\] \[\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\left(\mathbb{E}\bigg{[}d^{c}+ \left\|X_{T}^{d,0,K,\varepsilon,t,x}\right\|^{2}\bigg{]}\right)^{\frac{1}{2}}\] \[\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}\left((d^{c}+\|x\|^{2})e^{7cT} \right)^{\frac{1}{2}}\] \[=2(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}e^{3.5cT}.\]
This, (120), (123), the fact that \(c\geq 1\), Jensen's inequality, (124) show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\)\(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\begin{split}&\left|u^{d,K,\varepsilon}(t,x)\right|\\ &\leq\mathbb{E}\Big{[}\Big{|}g_{\varepsilon}^{d}(X_{T}^{d,0,K, \varepsilon,t,x})\Big{|}\Big{]}+\int_{t}^{T}\mathbb{E}\Big{[}\Big{|}f_{ \varepsilon}(u^{d,K,\varepsilon}(r,X_{r}^{d,0,K,\varepsilon,t,x}))\Big{|} \Big{]}\,dr\\ &\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}e^{ 3.5cT}+\int_{t}^{T}\Big{(}(cd^{c}T^{-3})^{\frac{1}{2}}+c^{\frac{1}{2}}\mathbb{ E}\Big{[}\Big{|}u^{d,K,\varepsilon}(r,X_{r}^{d,0,K,\varepsilon,t,x})\Big{|} \Big{]}\Big{)}\,dr\\ &\leq 3(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}e^{ 3.5cT}\\ &\quad+\int_{t}^{T}c\left[\sup_{y\in\mathbb{R}^{d}}\frac{\left|u ^{d,K,\varepsilon}(r,y)\right|}{e^{3.5c(T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}} \right]e^{3.5c(T-r)}\mathbb{E}\Bigg{[}\bigg{(}d^{c}+\left\|X_{r}^{d,0,K, \varepsilon,t,x}\right\|^{2}\bigg{)}^{\frac{1}{2}}\Bigg{]}\,dr\\ &\leq 3(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}e^{ 3.5cT}\\ &\quad+\int_{t}^{T}c\left[\sup_{y\in\mathbb{R}^{d}}\frac{\left|u ^{d,K,\varepsilon}(r,y)\right|}{e^{3.5c(T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}} \right]e^{3.5c(T-r)}\left((d^{c}+\|x\|^{2})e^{7c(r-t)}\right)^{\frac{1}{2}}\, dr\\ &=3(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}e^ {3.5cT}\\ &\quad+\int_{t}^{T}c\left[\sup_{y\in\mathbb{R}^{d}}\frac{\left|u ^{d,K,\varepsilon}(r,y)\right|}{e^{3.5c(T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}} \right]e^{3.5c(T-t)}(d^{c}+\|x\|^{2})^{\frac{1}{2}}\,dr.\end{split} \tag{126}\]

Dividing by \(e^{3.5c(T-t)}(d^{c}+\|x\|^{2})^{\frac{1}{2}}\) we then obtain that for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(\varepsilon\in(0,1)\) we have that

\[\sup_{y\in\mathbb{R}^{d}}\frac{\left|u^{d,K,\varepsilon}(t,y)\right|}{e^{3.5c (T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}\leq 3(cd^{c}T^{-1})^{\frac{1}{2}}e^{3.5cT}+ \int_{t}^{T}c\sup_{y\in\mathbb{R}^{d}}\frac{\left|u^{d,K,\varepsilon}(r,y) \right|}{e^{3.5c(T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}\,dr \tag{127}\]

This, (119), and Gronwall's inequality show for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\)\(\varepsilon\in(0,1)\) that

\[\sup_{y\in\mathbb{R}^{d}}\frac{\left|u^{d,K,\varepsilon}(t,y)\right|}{e^{3.5c (T-r)}(d^{c}+\|y\|^{2})^{\frac{1}{2}}}\leq 3(cd^{c}T^{-1})^{\frac{1}{2}}e^{3.5cT} \cdot e^{cT}. \tag{128}\]

This shows for all \(d,K\in\mathbb{N}\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\left|u^{d,K,\varepsilon}(t,x)\right|\leq 3(cd^{c}T^{-1})^{\frac{1}{2}}e^{8cT}(d^{c} +\|x\|^{2})^{\frac{1}{2}}. \tag{129}\]

Next, (81) and (122) show for all \(d\in\mathbb{N}\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\) that

\[\frac{T|f_{\varepsilon}(0)|+|g_{\varepsilon}^{d}(x)|}{(d^{c}+\|x\|^{2})^{ \frac{1}{2}}}\leq\frac{T\cdot T^{-\frac{3}{2}}c^{\frac{1}{2}}d^{\frac{1}{2}}+ 2(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}}{(d^{c}+\|x\|^{2} )^{\frac{1}{2}}}\leq 3T^{-\frac{1}{2}}(cd^{c})^{\frac{1}{2}} \tag{130}\]

This, [23, Corollary 3.12] (applied for all \(d,K\in\mathbb{N}\), \(\varepsilon\in(0,1)\) with \(f\gets f_{\varepsilon}\), \(g\gets g_{\varepsilon}^{d}\), \(\varphi\leftarrow(\mathbb{R}^{d}\ni x\mapsto(d^{c}+\|x\|^{2})^{\frac{1}{2}})\in(0,\infty)\), \((Y_{\cdot,}^{\theta}(\cdot))_{\theta\in\Theta}\leftarrow(X_{\cdot,}^{d,\theta, K,\varepsilon}\cdot)_{\theta\in\Theta}\leftarrow(U_{\cdot,}^{\theta})_{ \theta\in\Theta,m\in\mathbb{Z}}\leftarrow(U_{n,m}^{d,K,\varepsilon})_{\theta \in\Theta,n,m\in\mathbb{Z}}\) in the
notation of [23, Corollary 3.12]), (124), and (129) show for all \(d,K,n,m\in\mathds{N}\), \(\varepsilon\in(0,1)\), \(\theta\in\Theta\) that

\[\begin{split}&\sup_{t\in[0,T],x\in\mathds{R}^{d}}\frac{\mathbb{E} \bigg{[}\bigg{(}\Big{|}U_{n,m}^{d,\theta,K,\varepsilon}(t,x)-u^{d,K, \varepsilon}(t,x)\Big{|}^{2}\bigg{]}\bigg{)}^{\frac{1}{2}}}{(d^{c}+\|x\|^{2} )^{\frac{1}{2}}}\\ &\leq 2e^{\frac{m}{2}}m^{-\frac{n}{2}}(1+2Tc)^{N-1}e^{3.5cT}\left( \sup_{x\in\mathds{R}^{d}}\frac{T|f_{\varepsilon}(0)|+|g_{\varepsilon}^{d}(x)| }{(d^{c}+\|x\|^{2})^{\frac{1}{2}}}+Tc\sup_{t\in[0,T],x\in\mathds{R}^{d}}\frac{ \big{|}u^{d,K,\varepsilon}(t,x)\big{|}}{(d^{c}+\|x\|^{2})^{\frac{1}{2}}}\right) \\ &\leq 2e^{\frac{m}{2}}m^{-\frac{n}{2}}(e^{2cT})^{N-1}e^{3.5cT} \left(3T^{-\frac{1}{2}}(cd^{c})^{\frac{1}{2}}+Tc\cdot 3(cd^{c}T^{-1})^{\frac{1}{2}}e^{8cT }\right)\\ &\leq 2e^{\frac{m}{2}}m^{-\frac{n}{2}}(e^{2cT})^{N-1}e^{3.5cT}(1+Tc) \cdot 3(cd^{c}T^{-1})^{\frac{1}{2}}e^{8cT}\\ &\leq 6e^{\frac{m}{2}}m^{-\frac{n}{2}}(e^{2cT})^{N-1}e^{12cT}(cd^{c}T ^{-1})^{\frac{1}{2}}\\ &\leq 6e^{\frac{m}{2}}m^{-\frac{n}{2}}e^{12cTn}(cd^{c}T^{-1})^{ \frac{1}{2}}.\end{split} \tag{131}\]

This shows for all \(\theta\in\Theta\), \(d,K,n,m\in\mathds{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x\in\mathds{R}^{d}\) that

\[\mathbb{E}\bigg{[}\bigg{(}\Big{|}U_{n,m}^{d,\theta,K,\varepsilon}(t,x)-u^{d,K, \varepsilon}(t,x)\Big{|}^{2}\bigg{]}\bigg{)}^{\frac{1}{2}}\leq 6e^{\frac{m}{2}}m^{- \frac{n}{2}}e^{12cTn}(cd^{c}T^{-1})^{\frac{1}{2}}\left(d^{c}+\|x\|^{2}\right) ^{\frac{1}{2}}. \tag{132}\]

This completes the proof of Lemma 3.3. 

## 4. DNNs

### Properties of operations associated to DNNs

In Setting 4.1 below we introduce operations which are important for constructing the random DNN that represents the MLP approximations in the proof of Lemma 4.12.

**Setting 4.1**.: _Assume Setting 1.1, let \(\mathfrak{n}_{n}^{d}\in\mathbf{D}\), \(n\in[3,\infty)\cap\mathbb{Z}\), \(d\in\mathds{N}\), satisfy for all \(n\in[3,\infty)\cap\mathds{N}\), \(d\in\mathds{N}\) that_

\[\mathfrak{n}_{n}^{d}=(d,\underbrace{2d,\ldots,2d}_{(n-2)\text{ times}},d)\in\mathds{N}^{n}, \tag{133}\]

_let \(\mathfrak{n}_{n}\in\mathbf{D}\), \(n\in[3,\infty)\), satisfy for all \(n\in[3,\infty)\) that \(\mathfrak{n}_{n}=\mathfrak{n}_{n}^{1}\), let \(\boxplus\colon\mathbf{D}\times\mathbf{D}\to\mathbf{D}\) satisfy for all \(H\in\mathds{N}\), \(\alpha=(\alpha_{0},\alpha_{1},\ldots,\alpha_{H},\alpha_{H+1})\in\mathds{N}^{H +2}\), \(\beta=(\beta_{0},\beta_{1},\beta_{2},\ldots,\beta_{H},\beta_{H+1})\in\mathds{ N}^{H+2}\) that \(\alpha\boxplus\beta=(\alpha_{0},\alpha_{1}+\beta_{1},\ldots,\alpha_{H}+\beta_{H}, \beta_{H+1})\in\mathds{N}^{H+2},\) and let \(\odot\colon\mathbf{D}\times\mathbf{D}\to\mathbf{D}\) satisfy for all \(H_{1},H_{2}\in\mathds{N}\), \(\alpha=(\alpha_{0},\alpha_{1},\ldots,\alpha_{H_{1}},\alpha_{H_{1}+1})\in \mathds{N}^{H_{1}+2}\), \(\beta=(\beta_{0},\beta_{1},\ldots,\beta_{H_{2}},\beta_{H_{2}+1})\in\mathds{N }^{H_{2}+2}\) that \(\alpha\odot\beta=(\beta_{0},\beta_{1},\ldots,\beta_{H_{2}},\beta_{H_{2}+1}+ \alpha_{0},\alpha_{1},\alpha_{2},\ldots,\alpha_{H_{1}+1})\in\mathds{N}^{H_{1} +H_{2}+3}.\)_

To prove our main result in this section presented in Lemma 4.12 we employ several results presented in Lemmas 4.2-4.10, which are basic facts on DNNs. The proof of Lemmas 4.2-4.9 can be found in [8, 24] and therefore omitted.

**Lemma 4.2** (\(\odot\) is associative-[24, Lemma 3.3]).: _Assume Setting 4.1 and let \(\alpha,\beta,\gamma\in\mathbf{D}\). Then we have that \((\alpha\odot\beta)\odot\gamma=\alpha\odot(\beta\odot\gamma)\)._

**Lemma 4.3** (\(\boxplus\) and associativity-[24, Lemma 3.4]).: _Assume Setting 4.1, let \(H,k,l\in\mathds{N}\), and let \(\alpha,\beta,\gamma\in\big{(}\{k\}\times\mathds{N}^{H}\times\{l\}\big{)}\). Then_

1. _we have that_ \(\alpha\boxplus\beta\in\big{(}\{k\}\times\mathds{N}^{H}\times\{l\}\big{)}\)_,_
2. _we have that_ \(\beta\boxplus\gamma\in\big{(}\{k\}\times\mathds{N}^{H}\times\{l\}\big{)}\)_, and_
3. _we have that_ \((\alpha\boxplus\beta)\boxplus\gamma=\alpha\boxplus(\beta\boxplus\gamma)\)_._

**Lemma 4.4** (Triangle inequality-[24, Lemma 3.5]).: _Consider the notations given in Subsection 1.4, assume Setting 4.1, let \(k,l,H\in\mathds{N}\), \(\alpha,\beta\in\{k\}\times\mathds{N}^{H}\times\{l\}\). Then we have that \(\|\alpha\boxplus\beta\|\leq\|\alpha\|+\|\beta\|\)._

**Lemma 4.5** (DNNs for affine transformations-[24, Lemma 3.7]).: _Assume Setting 1.1 and let \(d,m\in\mathds{N}\), \(\lambda\in\mathds{R}\), \(b\in\mathds{R}^{d}\), \(a\in\mathds{R}^{m}\), \(\Psi\in\mathbf{N}\) satisfy that \(\mathcal{R}(\Psi)\in C(\mathds{R}^{d},\mathds{R}^{m})\). Then we have that \(\lambda\left((\mathcal{R}(\Psi))(\cdot+b)+a\right)\in\mathcal{R}(\{\Phi\in \mathds{N}\colon\mathcal{D}(\Phi)=\mathcal{D}(\Psi)\})\)._

**Lemma 4.6** (Composition of functions generated by DNNs-[24, Lemma 3.8]).: _Assume Setting 4.1 and let \(d_{1},d_{2},d_{3}\in\mathds{N}\), \(f_{1}\in C(\mathds{R}^{d_{2}},\mathds{R}^{d_{3}})\), \(f_{2}\in C(\mathds{R}^{d_{1}},\mathds{R}^{d_{2}})\), \(\alpha,\beta\in\mathbf{D}\) satisfy both that \(f_{1}\in\mathcal{R}(\{\Phi\in\mathds{N}\colon\mathcal{D}(\Phi)=\alpha\})\) as well as \(f_{2}\in\mathcal{R}(\{\Phi\in\mathds{N}\colon\mathcal{D}(\Phi)=\beta\})\). Then we have that \((f_{1}\circ f_{2})\in\mathcal{R}(\{\Phi\in\mathds{N}\colon\mathcal{D}(\Phi)= \alpha\odot\beta\})\)._
**Lemma 4.7** (Sum of DNNs of the same length-[24, Lemma 3.9]).: _Consider the notations given in Subsection 1.4, assume Setting 4.1 and let \(p,q,M,H\in\mathbb{N}\), \(\alpha_{1},\alpha_{2},\ldots,\alpha_{M}\in\mathbb{R}\), \(k_{i}\in\mathbf{D}\), \(g_{i}\in C(\mathds{R}^{p},\mathds{R}^{q})\), \(i\in[1,M]\cap\mathbb{N}\), satisfy for all \(i\in[1,M]\cap\mathbb{N}\) that \(\dim(k_{i})=H+2\) and \(g_{i}\in\mathcal{R}(\{\Phi\in\mathbb{N}\colon\mathcal{D}(\Phi)=k_{i}\})\). Then we have that \(\sum_{i=1}^{M}\alpha_{i}g_{i}\in\mathcal{R}\left(\left\{\Phi\in\mathbb{N} \colon\mathcal{D}(\Phi)=\mathbb{H}_{i}^{M}k_{i}\right\}\right)\)._

**Lemma 4.8** (Existence of DNNs with \(H\) hidden layers for \(\mathrm{Id}_{\mathbb{R}}\)\({}^{d}\)-[8, Lemma 3.6]).: _Assume Setting 4.1 and let \(d,H\in\mathbb{N}\). Then we have that \(\mathrm{Id}_{\mathbb{R}^{d}}\in\mathcal{R}(\{\Phi\in\mathbb{N}\colon\mathcal{ D}(\Phi)=\mathbb{H}_{H+2}^{d}\})\)._

**Lemma 4.9** ([8, Lemma 3.7]).: _Consider the notations given in Subsection 1.4, assume Setting 1.1, let \(H,p,q\in\mathbb{N}\), and let \(g\in C(\mathds{R}^{p},\mathds{R}^{q})\) satisfy that \(g\in\mathcal{R}(\{\Phi\in\mathbb{N}\colon\dim(\mathcal{D}(\Phi))=H+2\})\). Then for all \(n\in\mathbb{N}_{0}\) we have that \(g\in\mathcal{R}(\{\Phi\in\mathbb{N}\colon\dim(\mathcal{D}(\Phi))=H+2+n\})\)._

**Lemma 4.10**.: _Consider the notations given in Subsection 1.4 and assume Setting 4.1. Then for all \(n\in\mathbb{N}\), \(d_{0},d_{1},\ldots,d_{n}\in\mathbb{N}\), \(f_{1}\in C(\mathds{R}^{d_{1}},\mathds{R}^{d_{0}}),f_{2}\in C(\mathds{R}^{d_{2}},\mathds{R}^{d_{1}}),\ldots,f_{n}\in C(\mathds{R}^{d_{n}},\mathds{R}^{d_{n-1}})\), \(\phi_{1},\phi_{2},\ldots,\phi_{n}\in\mathbb{N}\) with \(\forall i\in[1,n]\cap\mathbb{Z}\colon f_{i}=\mathcal{R}(\phi_{i})\) we have that_

\[\left|\left|\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!
\(d\in\mathbb{N}\), \(\theta\in\Theta\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\), \(t\in[0,T)\) let \((X_{s}^{d,\theta,K,\varepsilon,t,x})_{s\in[t,T]}\) satisfy that \(X_{t}^{d,\theta,K,\varepsilon,t,x}=x\) and_

\[\begin{split} X_{s}^{d,\theta,K,\varepsilon,t,x}&=x+ \int_{t}^{s}\beta_{\varepsilon}^{d}(X_{\max\{t,\lfloor u-\rfloor\}}^{d,\theta, K,\varepsilon,t,x})du+\int_{t}^{s}\sigma_{\varepsilon}^{d}(X_{\max\{t,\lfloor u-\rfloor\}}^{d, \theta,K,\varepsilon,t,x})dW_{u}^{d,\theta}\\ &\quad+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}}\gamma_{ \varepsilon}^{d}(X_{\max\{t,\lfloor u-\rfloor\}}^{d,\theta,K,\varepsilon,t,x} )\,\tilde{N}^{d,\theta}(du,dz).\end{split} \tag{138}\]

_and let \(\omega\in\Omega\). Then there exists \((\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon})_{d\in\mathbb{N},\,\theta\in \Theta,\varepsilon\in(0,1),t\in[0,T),s\in(t,T]}\subseteq\mathbb{N}\) such that_

1. _for all_ \(d\in\mathbb{N}\)_,_ \(\theta\in\Theta\)_,_ \(\varepsilon\in(0,1)\)_,_ \(t\in[0,T)\)_,_ \(s\in(t,T]\)_,_ \(x\in\mathbb{R}^{d}\) _we have that_ \(\mathcal{R}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon})\in C(\mathbb{R}^{d}, \mathbb{R}^{d})\) _and_ \((\mathcal{R}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon}))(x)=X_{s}^{d,\theta, K,\varepsilon,t,x}(\omega)\)_,_
2. _for all_ \(d\in\mathbb{N}\)_,_ \(\theta\in\Theta\)_,_ \(\varepsilon\in(0,1)\)_,_ \(t_{1}\in[0,T)\)_,_ \(s_{1}\in(t_{1},T]\)_,_ \(t_{2}\in[0,T)\)_,_ \(s_{2}\in(t_{2},T]\)_,_ \(x\in\mathbb{R}^{d}\) _we have that_ \(\mathcal{D}(\mathcal{X}_{t,s_{1}}^{d,\theta,K,\varepsilon})=\mathcal{D}( \mathcal{X}_{t_{2},s_{2}}^{d,\theta_{2},K,\varepsilon})\)_,_
3. _for all_ \(d\in\mathbb{N}\)_,_ \(\theta\in\Theta\)_,_ \(\varepsilon\in(0,1)\)_,_ \(t\in[0,T)\)_,_ \(s\in(t,T]\) _we have that_ \(\dim(\mathcal{D}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon}))=K(\max\{\dim( \mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}})),\dim(\mathcal{D}(\Phi_{\sigma_{ \varepsilon}^{d},0})),\dim(\mathcal{D}(\Phi_{F_{\varepsilon}^{d},0}))\})-1)+1\)_, and_
4. _for all_ \(d\in\mathbb{N}\)_,_ \(\theta\in\Theta\)_,_ \(\varepsilon\in(0,1)\)_,_ \(t\in[0,T)\)_,_ \(s\in(t,T]\) _we have that_ \(\left|\kern-1.075pt\left|\kern-1.075pt\left|\mathcal{D}(\Phi_{\beta_{ \varepsilon}^{d}})\right|\kern-1.075pt\right|\kern-1.075pt\right|+\left|\kern-1.075pt \left|\kern-1.075pt\left|\mathcal{D}(\Phi_{F_{\varepsilon}^{d},0})\right| \kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt\right|+\left| \kern-1.075pt\left|\kern-1.075pt\left|\mathcal{D}(\Phi_{F_{\varepsilon}^{d},0}) \right|\kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt\)_._

Proof of Lemma 4.11.: Throughout this proof let the notation in Setting 4.1 be given. First, observe that for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\), \(k\in[1,K]\cap\mathbb{Z}\), \(t\in[0,T)\), \(s\in[\frac{kT}{K},\frac{(k+1)T}{K}]\) we have that

\[\begin{split} X_{s}^{d,\theta,K,\varepsilon,t,x}(\omega)& =X_{\max\{t,\frac{kT}{K}\}}^{d,\theta,K,\varepsilon,t,x}(\omega)+ \beta_{\varepsilon}^{d}\left(X_{\max\{t,\frac{kT}{K}\}}^{d,\theta,K, \varepsilon,t,x}(\omega)\right)\left(s-\max\{t,\frac{kT}{K}\}\right)\\ &\quad+\sigma_{\varepsilon}^{d}\left(X_{\max\{t,\frac{kT}{K}\}}^{d, \theta,K,\varepsilon,t,x}(\omega)\right)\left(W_{s}^{d,\theta}(\omega)-W_{ \max\{t,\frac{kT}{K}\}}^{d,\theta}(\omega)\right)\\ &\quad+F_{\varepsilon}^{d}\left(X_{\max\{t,\frac{kT}{K}\}}^{d, \theta,K,\varepsilon,t,x}(\omega)\right)\int_{\max\{t,\frac{kT}{K}\}}^{s} \int_{\mathbb{R}^{d}\setminus\{0\}}G^{d}(z)\tilde{N}^{d,\theta}(\omega)(du,dz). \end{split} \tag{139}\]

Next, for every \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(x\in\mathbb{R}^{d}\), \(\varepsilon\in(0,1)\), \(k\in[1,K]\cap\mathbb{Z}\), \(t\in[0,T)\), \(s\in(t,T]\) let \(J_{k}(s)\in\mathbb{R}\), \(\phi_{t,s,k}^{d,\theta,K,\varepsilon}(x)\in\mathbb{R}^{d}\) satisfy that

\[\begin{split} J_{k}(s)&=\max\{t,\frac{(k-1)T}{K}\} \mathbbm{1}_{[0,\max\{t,\frac{(k-1)T}{K}\}]}(s)\\ &\quad+s\mathbbm{1}_{(\max\{t,\frac{(k-1)T}{K}\},\max\{t,\frac{ kT}{K}\}]}(s)+\max\{t,\frac{kT}{K}\}\mathbbm{1}_{(\max\{t,\frac{kT}{K}\},T]}(s) \end{split} \tag{140}\]

and

\[\begin{split}\phi_{t,s,k}^{d,\theta,K,\varepsilon}(x)& =x+\beta_{\varepsilon}^{d}(x)\left(J_{k}(s)-\max\{t,\frac{(k-1)T}{K}\} \right)+\sigma_{\varepsilon}^{d}(x)\left(W_{J_{k}(s)}^{d,\theta}(\omega)-W_{ \max\{t,\frac{(k-1)T}{K}\}}^{d,\theta}(\omega)\right)\\ &\quad+F_{\varepsilon}^{d}(x)\int_{\max\{t,\frac{(k-1)T}{K}\}}^{s} \int_{\mathbb{R}^{d}\setminus\{0\}}G^{d}(z)\tilde{N}^{d,\theta}(\omega)(du,dz) \end{split} \tag{141}\]

Next, for every \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(k\in[1,K]\cap\mathbb{Z}\), \(t\in[0,T)\), \(s\in(t,T]\) let

\[\begin{split}\psi_{t,s,k}^{d,\theta,K,\varepsilon}=\phi_{t,s,k}^{d, \theta,K,\varepsilon}\circ\phi_{t,s,k-1}^{d,\theta,K,\varepsilon}\circ\ldots \circ\phi_{t,s,1}^{d,\theta,K,\varepsilon}.\end{split} \tag{142}\]

Note that for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(k\in[1,K-1]\cap\mathbb{Z}\), \(s\in[0,\max\{t,\frac{(k-1)T}{K}\}]\) we have that \(\phi_{t,s,k}^{d,\theta,K,\varepsilon}=\mathrm{Id}_{\mathrm{R}^{d}}\). This ensures for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(k\in[1,K-1]\cap\mathbb{Z}\), \(n\in[k+1,K]\cap\mathbb{Z}\), \(s\in[0,\max\{t,\frac{kT}{K}\}]\) that \(\psi_{t,s,k}^{d,\theta,K,\varepsilon}=\psi_{t,s,n}^{d,\theta,K,\varepsilon}\) and in particular \(\psi_{t,s,k}^{d,\theta,K,\varepsilon}=\psi_{t,s,K}^{d,\theta,K,\varepsilon}\). Observe that for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(k\in[1,K]\cap\mathbb{Z}\), \(s\in[0,\
Lemma 4.8 shows for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,\infty)\) that

\[\mathrm{Id}_{\mathbb{R}^{d}}\in\mathcal{R}(\{\Phi\in\mathbf{N}\colon\mathcal{D}( \Phi)=\mathfrak{n}_{\dim(\Phi_{\beta_{\varepsilon}^{d}})}^{d}\}) \tag{145}\]

This, (141), (144), and Lemma 4.7 show for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\delta,\varepsilon\in(0,1)\), \(k\in[1,K]\cap\mathbb{Z}\), \(t\in[0,T)\), \(s\in(t,T]\) that

\[\phi_{t,s,k}^{d,\theta,K,\varepsilon}(\cdot)\in\mathcal{R}\left(\left\{\Phi \in\mathbf{N}\colon\mathcal{D}(\Phi)=\mathfrak{n}_{\dim(\Phi_{\beta_{ \varepsilon}^{d}})}^{d}\boxplus\mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}}) \boxplus\mathcal{D}(\Phi_{\sigma_{\varepsilon}^{d},0})\boxplus\mathcal{D}( \Phi_{F_{\varepsilon}^{d},0})\right\}\right) \tag{146}\]

This, (143), and Lemma 4.6 show that there exists \((\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon})_{d\in\mathbb{N},\theta\in\Theta, \varepsilon\in(0,1),t\in[0,T),s\in(t,T]}\subseteq\mathbf{N}\) such that for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(t\in[0,T)\), \(s\in(t,T]\), \(x\in\mathbb{R}^{d}\) we have that

\[\mathcal{D}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon})=\overset{K}{\underset{k =1}{\overset{\odot}{\sim}}}\left[\mathfrak{n}_{\dim(\Phi_{\beta_{\varepsilon} ^{d}})}^{d}\boxplus\mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}})\boxplus\mathcal{ D}(\Phi_{\sigma_{\varepsilon}^{d},0})\boxplus\mathcal{D}(\Phi_{F_{\varepsilon}^{d},0}) \right], \tag{147}\] \[(\mathcal{R}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon}))(x)=X_{s} ^{d,\theta,K,\varepsilon,t,x}(\omega).\]

This, the definition of \(\odot\) and an induction argument show that for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(t\in[0,T)\), \(s\in(t,T]\), \(x\in\mathbb{R}^{d}\) we have that

\[\dim(\mathcal{D}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon}))=K(\dim( \mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}}))-1)+1. \tag{148}\]

Next, (147), Lemma 4.10, the triangle inequality (cf. Lemma 4.4), and (133) show that for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(t\in[0,T)\), \(s\in(t,T]\), \(x\in\mathbb{R}^{d}\) we have that

\[\begin{split}\left|\left|\mathcal{D}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon})\right|\right|&=\left|\left|\overset{K}{\underset{k =1}{\overset{\odot}{\sim}}}\left[\mathfrak{n}_{\dim(\Phi_{\beta_{\varepsilon }^{d}})}^{d}\boxplus\mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}})\boxplus \mathcal{D}(\Phi_{\sigma_{\varepsilon}^{d},0})\boxplus\mathcal{D}(\Phi_{F_{ \varepsilon}^{d},0})\right]\right|\right|\\ &\leq\max\left\{2d,\left|\left|\mathfrak{n}_{\dim(\Phi_{\beta_{ \varepsilon}^{d}})}^{d}\boxplus\mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}}) \boxplus\mathcal{D}(\Phi_{\sigma_{\varepsilon}^{d},0})\boxplus\mathcal{D}( \Phi_{F_{\varepsilon}^{d},0})\right|\right|\right\}\\ &\leq\max\left\{2d,\left|\left|\mathfrak{n}_{\dim(\Phi_{\beta_{ \varepsilon}^{d}})}^{d}\right|\right|+\left|\left|\mathcal{D}(\Phi_{\beta_{ \varepsilon}^{d}})\right|\right|+\left|\left|\mathcal{D}(\Phi_{\sigma_{ \varepsilon}^{d},0})\right|\right|+\left|\left|\mathcal{D}(\Phi_{F_{\varepsilon }^{d},0})\right|\right|\right\}\\ &=2d+\left|\left|\mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}}) \right|\right|+\left|\left|\mathcal{D}(\Phi_{\sigma_{\varepsilon}^{d},0}) \right|\right|+\left|\left|\mathcal{D}(\Phi_{F_{\varepsilon}^{d},0})\right| \right|\right|\end{split} \tag{149}\]

The proof of Lemma 4.11 is thus completed. 

### DNN representation of our MLP approximations

In Lemma 4.12 below we prove that the MLP approximations under consideration can be represented by DNNs.

**Lemma 4.12**.: _Assume the setting of Lemma 4.11, for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(f_{\varepsilon}\in C(\mathbb{R},\mathbb{R})\), \(g_{\varepsilon}^{d}\in C(\mathbb{R}^{d},\mathbb{R})\), \(\Phi_{f_{\varepsilon}},\Phi_{g_{\varepsilon}^{d}}\in\mathbf{N}\) satisfy that \(\mathcal{R}(\Phi_{f_{\varepsilon}})=f_{\varepsilon}\) and \(\mathcal{R}(\Phi_{g_{\varepsilon}^{d}})=g_{\varepsilon}^{d}\), let \(\mathfrak{t}^{\theta}\colon\Omega\to[0,1]\), \(\theta\in\Theta\), be i.i.d random variables which satisfy for all \(t\in(0,1)\) that \(\mathbb{P}(\mathfrak{t}^{0}\leq t)=t\), for every \(\theta\in\Theta\), \(t\in[0,T]\) let \(\mathfrak{z}_{t}^{\theta}\colon\Omega\to\mathbb{R}\) satisfy for all \(\theta\in\Theta\) that \(\mathfrak{z}_{t}^{\theta}=t+(T-t)\mathfrak{t}^{\theta}\), assume for all \(d\in\mathbb{N}\) that \((\mathfrak{t}^{\theta})_{\theta\in\Theta}\), \((N^{d,\theta})_{\theta\in\Theta}\) and \((W^{d,\theta})_{\theta\in\Theta}\), are independent, for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(U_{n,m}^{d,\theta,K,\varepsilon}\colon[0,T]\times\mathbb{R}^{d}\times\Omega\to \mathbb{R}\), \(\theta\in\Theta\), \(n,m\in\mathbb{Z}\), satisfy for all \(\theta\in\Theta\), \(n\in\mathbb{N}_{0}\), \(m\in\mathbb{N}\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) that_

\[\begin{split}& U_{n,m}^{d,\theta,K,\varepsilon}(t,x)=\frac{ \mathbbm{1}_{\mathbb{N}}(n)}{m^{n}}\sum_{i=1}^{m^{n}}g_{\varepsilon}^{d} \Big{(}X_{T}^{d,(\theta,0,-i),K,\varepsilon,t,x}\Big{)}\\ &\quad+\sum_{\ell=0}^{n-1}\frac{(T-t)}{m^{n-\ell}}\sum_{i=1}^{m^{ n-\ell}}\Big{(}f_{\varepsilon}\circ U_{\ell,m}^{d,(\theta,\ell,i),K,\varepsilon}- \mathbbm{1}_{\mathbb{N}}(\ell)f_{\varepsilon}\circ U_{\ell-1,m}^{d,(\theta,- \ell,i),K,\varepsilon}\Big{)}\Big{(}\mathfrak{z}_{t}^{(\theta,\ell,i)},X_{ \mathfrak{z}_{t}^{(\theta,\ell,i)}}^{d,(\theta,\ell,i),K,\varepsilon,t,x} \Big{)},\end{split} \tag{150}\]

_and let \((c_{d,\varepsilon})_{d\in\mathbb{N},\varepsilon\in(0,1)}\subseteq\mathbb{R}\) satisfy for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) that_

\[c_{d,\varepsilon}\geq 2d+\left|\mathcal{D}(\Phi_{f_{\varepsilon}})\right|\right|+ \left|\left|\mathcal{D}(\Phi_{g_{\varepsilon}^{d}})\right|\right|+\left| \left|\mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}})\right|\right|+\left|\left| \mathcal{D}(\Phi_{\sigma_{\varepsilon}^{d},0})\right|\right|+\left|\left| \mathcal{D}(\Phi_{F_{\varepsilon}^{d},0})\right|\right|\right|\]. (151)

_Then for all \(m\in\mathbb{N}\), \(n\in\mathbb{N}_{0}\), \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) there exists \((\Phi_{n,m,t}^{d,K,\varepsilon})_{t\in[0,T],\theta\in\Theta}\subseteq\mathbf{N}\) such that_

1. _we have for all_ \(t_{1},t_{2}\in[0,T]\)_,_ \(\theta_{1},\theta_{2}\in\Theta\) _that_ \(\mathcal{D}(\Phi_{n,m,t_{1}}^{d,\theta_{1},K,\varepsilon}
_,_
2. _we have for all_ \(t\in[0,T]\)_,_ \(\theta\in\Theta\) _that_ \[\begin{split}\dim(\mathcal{D}(\Phi_{n,m,t}^{d,\theta,K,\varepsilon})) &=(n+1)\left[K\left(\max\left\{\dim(\mathcal{D}(\Phi_{\beta^{d}_{ \varepsilon}})),\dim(\mathcal{D}(\Phi_{\sigma^{d}_{\varepsilon}}))\right\}-1 \right)+1\right]\\ &\qquad+n(\dim(\mathcal{D}(\Phi_{f_{\varepsilon}}))-2)+\dim( \mathcal{D}(\Phi_{g^{d}_{\varepsilon}}))-1,\end{split}\] (152)
3. _we have for all_ \(t\in[0,T]\)_,_ \(\theta\in\Theta\) _that_ \(\left\|\kern-1.075pt\left|\mathcal{D}(\Phi_{n,m,t}^{d,\theta,K,\varepsilon}) \right|\kern-1.075pt\right|\kern-1.075pt\right|\leq c_{d,\varepsilon}(3m)^{n}\)_, and_
4. _we have for all_ \(t\in[0,T]\)_,_ \(\theta\in\Theta\)_,_ \(x\in\mathbb{R}^{d}\) _that_ \(U_{n,m}^{d,\theta,K,\varepsilon}(t,x,\omega)=(\mathcal{R}(\Phi_{n,m,t}^{d, \theta,K,\varepsilon}))(x)\)_._

Proof of Lemma 4.12.: Throughout this proof let the notation in Setting 4.1 be given. First, Lemma 4.11 and (151) show that there exists \((\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon})_{d\in\mathbb{N},\theta\in\Theta, \varepsilon\in(0,1),t\in[0,T),s\in(t,T]}\subseteq\mathbf{N}\) such that

for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(t\in[0,T)\), \(s\in(t,T]\), \(x\in\mathbb{R}^{d}\) we have that \[\mathcal{R}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon}))(x)=X_{s}^{d,\theta,K, \varepsilon,t,x}(\omega),\] (153) for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(t_{1}\in[0,T)\), \(s_{1}\in(t_{1},T]\), \(t_{2}\in[0,T)\), \(s_{2}\in(t_{2},T]\), \(x\in\mathbb{R}^{d}\) we have that

\[\mathcal{D}(\mathcal{X}_{t_{1},s_{1}}^{d,\theta_{1},K,\varepsilon})=\mathcal{ D}(\mathcal{X}_{t_{2},s_{2}}^{d,\theta_{2},K,\varepsilon})\] (154) for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(t\in[0,T)\), \(s\in(t,T]\) we have that \[\dim(\mathcal{D}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon}))=K(\max\{\dim( \mathcal{D}(\Phi_{\beta^{d}_{\varepsilon}})),\dim(\mathcal{D}(\Phi_{\sigma^{ d}_{\varepsilon},0})),\dim(\mathcal{D}(\Phi_{F^{d}_{\varepsilon},0}))\}-1)+1,\] (155) and for all \(d\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(t\in[0,T)\), \(s\in(t,T]\) we have that \[\left|\kern-1.075pt\left|\kern-1.075pt\left|\mathcal{D}(\mathcal{X}_{t,s}^{d, \theta,K,\varepsilon})\right|\kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt \right|\leq 2d+\left|\kern-1.075pt\left|\kern-1.075pt\left|\mathcal{D}(\Phi_{ \beta^{d}_{\varepsilon}})\right|\kern-1.075pt\right|\kern-1.075pt\right|+\left| \kern-1.075pt\left|\mathcal{D}(\Phi_{F^{d}_{\varepsilon},0})\right|\kern-1.075pt \right|\kern-1.075pt\right|\kern-1.075pt\right|\leq c_{d,\varepsilon}. \tag{156}\]

Throughout the rest of this proof let \(m\in\mathbb{N}\), \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) be fixed, let \(\mathrm{Id}_{\mathbb{R}}\) be the identity on \(\mathbb{R}\), and let \(L\in\mathbb{Z}\) satisfy that

\[L=\dim(\mathcal{D}(\mathcal{X}_{t,s}^{d,\theta,K,\varepsilon}))=K\left(\max \left\{\dim(\mathcal{D}(\Phi_{\beta^{d}_{\varepsilon}})),\dim(\mathcal{D}( \Phi_{\sigma^{d}_{\varepsilon}}))\right\}-1\right)+1. \tag{157}\]

(cf. (155)). We will prove the lemma via induction on \(n\in\mathbb{N}_{0}\). First, the base case \(n=0\) is true since the \(0\)-function can be represented by DNN function of arbitrary length. For the induction step \(\mathbb{N}_{0}\ni n\mapsto n+1\in\mathbb{N}\) let \(n\in\mathbb{N}_{0}\) and assume that there exists \((\Phi_{\ell,m,t}^{d,\theta,K,\varepsilon})_{t\in[0,T],\theta\in\Theta}\subseteq \mathbf{N}\), \(\ell\in[0,n]\cap\mathbb{Z}\), such that

we have for all \(t_{1},t_{2}\in[0,T]\), \(\theta_{1},\theta_{2}\in\Theta\), \(\ell\in[0,n]\cap\mathbb{Z}\) that

\[\mathcal{D}(\Phi_{\ell,m,t_{1}}^{d,\theta_{1},K,\varepsilon})=\mathcal{D}(\Phi_ {\ell,m,t_{2}}^{d,\theta_{2},K,\varepsilon}), \tag{158}\]

we have for all \(t\in[0,T]\), \(\theta\in\Theta\), \(\ell\in[0,n]\cap\mathbb{Z}\) that

\[\dim(\mathcal{D}(\Phi_{\ell,m,t}^{d,\theta,K,\varepsilon}))=(\ell+1)L+\ell( \dim(\mathcal{D}(\Phi_{f_{\varepsilon}}))-2)+\dim(\mathcal{D}(\Phi_{g_{ \varepsilon}^{d}}))-1, \tag{159}\]

we have for all \(t\in[0,T]\), \(\theta\in\Theta\), \(\ell\in[0,n]\cap\mathbb{Z}\) that

\[\left|\kern-1.075pt\left|\kern-1.075pt\left|\Phi_{\ell,m,t}^{d,\theta,K, \varepsilon}\right|\kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt\right| \kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt\right| \kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt\right|\kern-1.075pt \tag{160}\]

and

we have for all \(t\in[0,T]\), \(\theta\in\Theta\), \(x\in\mathbb{R}^{d}\), \(\ell\in[0,n]\cap\mathbb{Z}\) that

\[U_{\ell,m}^{d,\theta,K,\varepsilon}(t,x,\omega)=(\mathcal{R}(\Phi_{\ell,m,t}^{d, \theta,K,\varepsilon}))(x). \tag{161}\]

Next, Lemma 4.8, the fact that \(g_{\varepsilon}^{d}=\mathcal{R}(\Phi_{g_{\varepsilon}^{d}})\), (153), (154), and Lemma 4.6 show for all \(\theta\in\Theta\), \(i\in[1,m^{n+1}]\cap\mathbb{Z}\), \(t\in[0,T]\) that

\[\begin{split}& g_{\varepsilon}^{d}\Big{(}X_{t,T}^{d,(\theta,0,-i),K, \varepsilon},(\omega)\Big{)}=\mathrm{Id}_{\mathbb{R}}\,\left(g_{ \varepsilon}^{d}\Big{(}X_{t,T}^{d,(\theta,0,-i),K,\varepsilon},(\omega)\Big{)} \right)\\ &\in\mathcal{R}\left(\left\{\Phi\in\mathbf{N}\colon\mathcal{D}( \Phi)=\mathfrak{n}_{(n+1)\big{(}\dim(\mathcal{D}(\Phi_{f_{\varepsilon}}))-2+L \big{)}+1}\odot\mathcal{D}(\Phi_{g_{\varepsilon}^{d}})\odot\mathcal{D}( \mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right\}\right).\end{split} \tag{162}\]
In addition, the definition of \(\odot\), (133), and (157) show that

\[\begin{split}&\dim\left(\mathfrak{n}_{(n+1)\big{(}\dim(\mathcal{D}( \Phi_{f_{\varepsilon}}))-2+L\big{)}+1}\odot\mathcal{D}(\Phi_{g_{\varepsilon}^{ d}})\odot\mathcal{D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right)\\ &=\dim\left(\mathfrak{n}_{(n+1)\big{(}\dim(\mathcal{D}(\Phi_{f_{ \varepsilon}}))-2+L\big{)}+1}\right)+\dim\left(\mathcal{D}(\Phi_{g^{d}}) \right)+\dim\left(\mathcal{D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right)-2 \\ &=(n+1)\big{(}\dim(\mathcal{D}(\Phi_{f_{\varepsilon}}))-2+L\big{)} +1+\dim\left(\mathcal{D}(\Phi_{g^{d}})\right)+L-2\\ &=(n+1)\big{(}\dim(\mathcal{D}(\Phi_{f_{\varepsilon}}))-2\big{)} +(n+2)L+\dim\left(\mathcal{D}(\Phi_{g^{d}})\right)-1.\end{split} \tag{163}\]

Next, Lemma 4.8, the fact that \(f_{\varepsilon}=\mathcal{R}(\Phi_{f_{\varepsilon}})\), (158), (161), (153), (154), and Lemma 4.6 show for all \(\theta\in\Theta\), \(i\in[1,m^{n+1}]\cap\mathbb{Z}\), \(t\in[0,T]\) that

\[\begin{split}&\Big{(}f_{\varepsilon}\circ U_{n,m}^{d,(\theta,n,i),K, \varepsilon}\Big{)}\Big{(}\mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega),X_{t, \mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega)}^{d,(\theta,\ell,i),K,\varepsilon, t}(\omega)\Big{)}\\ &\in\mathcal{R}\left(\Big{\{}\Phi\in\mathbf{N}\colon\mathcal{D}( \Phi)=\mathcal{D}(\Phi_{f_{\varepsilon}})\odot\mathcal{D}(\Phi_{n,m,0}^{d,0,K, \varepsilon})\odot\mathcal{D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\Big{\}} \right).\end{split} \tag{164}\]

In addition, the definition of \(\odot\), (159), and (157) show that

\[\begin{split}&\dim\left(\mathcal{D}(\Phi_{f_{\varepsilon}}) \odot\mathcal{D}(\Phi_{n,m,0}^{d,0,K,\varepsilon})\odot\mathcal{D}(\mathcal{X }_{0,T}^{d,0,K,\varepsilon})\right)\\ &=\dim\left(\mathcal{D}(\Phi_{f_{\varepsilon}})\right)+\dim \left(\mathcal{D}(\Phi_{n,m,0}^{d,0,K,\varepsilon})\right)+\dim\left(\mathcal{ D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right)-2\\ &=\dim\left(\mathcal{D}(\Phi_{f_{\varepsilon}})\right)+\Big{(}(n +1)L+n(\dim(\mathcal{D}(\Phi_{f_{\varepsilon}}))-2)+\dim(\mathcal{D}(\Phi_{g ^{d}_{\varepsilon}}))-1\Big{)}+L-2\\ &=(n+1)\big{(}\dim(\mathcal{D}(\Phi_{f_{\varepsilon}}))-2\big{)} +(n+2)L+\dim\left(\mathcal{D}(\Phi_{g^{d}_{\varepsilon}})\right)-1.\end{split} \tag{165}\]

Furthermore, the fact that \(f_{\varepsilon}=\mathcal{R}(\Phi_{f_{\varepsilon}})\), Lemma 4.8, (158), (161), (153), (154), and Lemma 4.6 show for all \(\theta\in\Theta\), \(i\in[1,m^{n+1}]\cap\mathbb{Z}\), \(t\in[0,T]\), \(\ell\in[0,n-1]\cap\mathbb{Z}\) that

\[\begin{split}&\Big{(}f_{\varepsilon}\circ U_{\ell,m}^{d,(\theta, \ell,i),K,\varepsilon}\Big{)}\Big{(}\mathfrak{T}_{t}^{(\theta,\ell,i)}( \omega),X_{\mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega)}^{d,(\theta,\ell,i),K, \varepsilon,t},\omega\Big{)}\\ &=\Big{(}f_{\varepsilon}\circ\mathrm{Id}_{\mathrm{R}}\circ U_{ \ell,m}^{d,(\theta,\ell,i),K,\varepsilon}\Big{)}\Big{(}\mathfrak{T}_{t}^{( \theta,\ell,i)}(\omega),X_{\mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega)}^{d,( \theta,\ell,i),K,\varepsilon,t},\omega\Big{)}\\ &\quad\in\mathcal{R}\left(\Big{\{}\Phi\in\mathbf{N}\colon \mathcal{D}(\Phi)=\mathcal{D}(\Phi_{f_{\varepsilon}})\odot\mathfrak{n}_{(n- \ell)\big{(}\mathcal{D}(\Phi_{f_{\varepsilon}})-2+L\big{)}+1}\odot\mathcal{D }(\Phi_{\ell,m,0}^{d,0,K,\varepsilon})\odot\mathcal{D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\Big{\}}\right).\end{split} \tag{166}\]

In addition, the definition of \(\odot\), (133), (159), and (157) show for all \(\ell\in[0,n-1]\cap\mathbb{Z}\) that

\[\begin{split}&\dim\left(\mathcal{D}(\Phi_{f_{\varepsilon}}) \odot\mathfrak{n}_{(n-\ell)\big{(}\mathcal{D}(\Phi_{f_{\varepsilon}})-2+L \big{)}+1}\odot\mathcal{D}(\Phi_{\ell,m,0}^{d,0,K,\varepsilon})\odot\mathcal{D }(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right)\\ &=\dim\left(\mathcal{D}(\Phi_{f_{\varepsilon}})\right)+\dim\left( \mathfrak{n}_{(n-\ell)\big{(}\mathcal{D}(\Phi_{f_{\varepsilon}})-2+L\big{)}+1} \right)+\dim\left(\mathcal{D}(\Phi_{\ell,m,0}^{d,0,K,\varepsilon})\right)+ \dim\left(\mathcal{D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right)-3\\ &=\dim\left(\mathcal{D}(\Phi_{f_{\varepsilon}})\right)+\left((n- \ell)\big{(}\mathcal{D}(\Phi_{f_{\varepsilon}})-2+L\big{)}+1\right)\\ &\quad+\Big{(}(\ell+1)L+\ell(\dim(\mathcal{D}(\Phi_{f_{\varepsilon}}) )-2)+\dim(\mathcal{D}(\Phi_{g^{d}_{\varepsilon}}))-1\Big{)}+L-3\\ &=(n+1)\big{(}\dim(\mathcal{D}(\Phi_{f_{\varepsilon}}))-2\big{)} +(n+2)L+\dim\left(\mathcal{D}(\Phi_{g^{d}_{\varepsilon}})\right)-1.\end{split} \tag{167}\]

Furthermore, the fact that \(f_{\varepsilon}=\mathcal{R}(\Phi_{f_{\varepsilon}})\), Lemma 4.8, (158), (161), (153), (154), and Lemma 4.6 show for all \(\theta\in\Theta\), \(i\in[1,m^{n+1}]\cap\mathbb{Z}\), \(t\in[0,T]\), \(\ell\in[1,n]\cap\mathbb{Z}\) that

\[\begin{split}&\Big{(}f_{\varepsilon}\circ U_{\ell,m}^{d,(\theta, \ell,i),K,\varepsilon}\Big{)}\Big{(}\mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega),X_{ \mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega)}^{d,(\theta,\ell,i),K,\varepsilon, t}(\omega),\omega\Big{)}\\ &=\Big{(}f_{\varepsilon}\circ\mathrm{Id}_{\mathrm{R}}\circ U_{ \ell-1,m}^{d,(\theta,\ell,i),K,\varepsilon}\Big{)}\Big{(}\mathfrak{T}_{t}^{( \theta,\ell,i)}(\omega),X_{\mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega)}^{d,(\theta, \ell,i),K,\varepsilon,t}(\omega),\omega\Big{)}\\ &\in\mathcal{R}\left(\Big{\{}\Phi\in\mathbf{N}\colon\mathcal{D}( \Phi)=\mathcal{D}(\Phi_{f_{\varepsilon}})\odot\mathfrak{n}_{(n-\ell+1) \big{(}\mathcal{D}(\Phi_{f_{\varepsilon}})-2+L\big{)}+1}\odot\mathcal{D}( \Phi_{\ell-1,m,0}^{d,0,K,\varepsilon})\odot\mathcal{D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\Big{\}}\right).\end{split} \tag{168}\]
In addition, the definition of \(\odot\), (133), (159), and (157) show for all \(\ell\in[1,n]\cap\mathbb{Z}\) that

\[\begin{split}&\dim\left(\mathcal{D}(\Phi_{f_{\ell}})\odot\mathfrak{ n}_{(n-\ell+1)\big{(}\mathcal{D}(\Phi_{f_{\ell}})-2+L\big{)}+1}\odot\mathcal{D}( \Phi_{\ell-1,m,0}^{d,0,K,\varepsilon})\odot\mathcal{D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right)\\ &=\dim\left(\mathcal{D}(\Phi_{f_{\ell}})\right)+\dim\left( \mathfrak{n}_{(n-\ell+1)\big{(}\mathcal{D}(\Phi_{f_{\ell}})-2+L\big{)}+1} \right)\\ &\quad+\dim\left(\mathcal{D}(\Phi_{\ell-1,m,0}^{d,0,K,\varepsilon })\right)+\dim\left(\mathcal{D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right)- 3\\ &=\dim\left(\mathcal{D}(\Phi_{f_{\ell}})\right)+\left((n-\ell+1) \big{(}\mathcal{D}(\Phi_{f_{\ell}})-2+L\big{)}+1\right)\\ &\quad+\left(\ell L+(\ell-1)(\dim(\mathcal{D}(\Phi_{f_{\ell}}))- 2)+\dim(\mathcal{D}(\Phi_{g_{\ell}^{d}}))-1\right)+L-3\\ &=(n+1)\big{(}\dim(\mathcal{D}(\Phi_{f_{\ell}}))-2\big{)}+(n+2)L +\dim\left(\mathcal{D}(\Phi_{g_{\ell}^{d}})\right)-1.\end{split} \tag{169}\]

Now, (162)-(169) and Lemma 4.7 show that there exists \((\Phi_{n+1,m,t}^{d,\theta,K,\varepsilon})_{t\in[0,T],\theta\in\Theta}\) such that \(t\in[0,T]\), \(\theta\in\Theta\), \(x\in\mathbb{R}^{d}\) we have that

\[\begin{split}(\mathcal{R}(\Phi_{n+1,m,t}^{d,\theta,K, \varepsilon}))(x)&=\frac{1}{m^{n+1}}\sum_{i=1}^{m^{n+1}}g_{ \varepsilon}^{d}\Big{(}X_{T}^{d,(\theta,0,-i),K,\varepsilon,t,x}(\omega) \Big{)}\\ &\quad+\frac{1}{m}\sum_{i=1}^{m}\Bigl{(}f_{\varepsilon}\circ U_{n,m}^{d,(\theta,n,i),K,\varepsilon}\Bigr{)}\Bigl{(}\mathfrak{T}_{t}^{(\theta, \ell,i)}(\omega),X_{\mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega)}^{d,(\theta, \ell,i),K,\varepsilon,t,x}(\omega),\omega\Bigr{)}\\ &\quad+\sum_{\ell=0}^{n-1}\frac{(T-t)}{m^{n+1-\ell}}\sum_{i=1}^{m ^{n+1-\ell}}\Bigl{(}f_{\varepsilon}\circ U_{\ell,m}^{d,(\theta,\ell,i),K, \varepsilon}\Bigr{)}\Bigl{(}\mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega),X_{ \mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega)}^{d,(\theta,\ell,i),K,\varepsilon, t,x}(\omega),\omega\Bigr{)}\\ &\quad-\sum_{\ell=1}^{n}\frac{(T-t)}{m^{n+1-\ell}}\sum_{i=1}^{m ^{n+1-\ell}}\Bigl{(}f_{\varepsilon}\circ U_{\ell-1,m}^{d,(\theta,-\ell,i),K, \varepsilon}\Bigr{)}\Bigl{(}\mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega),X_{ \mathfrak{T}_{t}^{(\theta,\ell,i)}(\omega)}^{d,(\theta,\ell,i),K,\varepsilon, t,x}(\omega),\omega\Bigr{)}\\ &=U_{n+1,m}^{d,\theta,K,\varepsilon}(t,x),\end{split} \tag{170}\]

\[\mathcal{D}(\Phi_{n+1,m,t}^{d,K,\varepsilon})=(n+1)\bigl{(}\dim(\mathcal{D}( \Phi_{f_{\ell}}))-2\bigr{)}+(n+2)L+\dim\left(\mathcal{D}(\Phi_{g_{\ell}^{d}}) \right)-1, \tag{171}\]

and

\[\begin{split}\mathcal{D}(\Phi_{n+1,m,t}^{d,\theta,K,\varepsilon })&=\left[\overset{m^{n+1}}{\underset{i=1}{\overset{n}{\equiv}}} \left[\mathfrak{n}_{(n+1)\big{(}\dim(\mathcal{D}(\Phi_{f_{\ell}}))-2+L\big{)} +1}\odot\mathcal{D}(\Phi_{g_{\ell}^{d}})\odot\mathcal{D}(\mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right]\right]\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
Next, Lemma 4.10, (151), (160), and (156) show that

\[\begin{split}&\left|\!\left|\!\left|\mathcal{D}(\Phi_{f_{\varepsilon}}) \odot\mathcal{D}(\Phi_{n,m,0}^{d,0,K,\varepsilon})\odot\mathcal{D}(\mathcal{ X}_{0,T}^{d,0,K,\varepsilon})\right|\!\right|\!\right|\\ &\leq\max\left\{2d,\left|\!\left|\!\left|\mathcal{D}(\Phi_{f_{ \varepsilon}})\right|\!\right|\!\right|\!\right|\!\left|\!\left|\!\left| \mathcal{D}(\Phi_{n,m,0}^{d,0,K,\varepsilon})\right|\!\right|\!\right|\! \right|\!\!\left|\!\right|\!\right|\!\!\left|\!\left|\!\left|\mathcal{D}( \mathcal{X}_{0,T}^{d,0,K,\varepsilon})\right|\!\right|\!\right|\!\right\}\leq c _{d,\varepsilon}(3m)^{n}.\end{split} \tag{175}\]

Furthermore, Lemma 4.10, (151), (160), and (156) show for all \(\ell\in[0,n-1]\cap\mathbb{Z}\) that

(176)

In addition, Lemma 4.10, (151), (160), and (156) show for all \(\ell\in[1,n]\cap\mathbb{Z}\) that

(177)

Now, (172), the triangle inequality, and (174)-(177) show for all \(t\in[0,T]\), \(\theta\in\Theta\) that

(178)

and

\[\begin{split}&\left|\!\left|\!\left|\mathcal{D}(\Phi_{n+1,m,t}^{ 4\theta,K,\varepsilon})\right|\!\right|\!\right|\\ &\leq\left[\sum_{i=1}^{m+1}c_{d,\varepsilon}\right]+\left[\sum_{ i=1}^{m}c_{d,\varepsilon}(3m)^{n}\right]+\left[\sum_{\ell=0}^{n-1}\sum_{i=1}^{m+1- \ell}c_{d,\varepsilon}(3m)^{\ell}\right]+\left[\sum_{\ell=1}^{n}\sum_{i=1}^{m ^{n+1-\ell}}c_{d,\varepsilon}(3m)^{\ell-1}\right]\\ &=m^{n+1}c_{d,\varepsilon}+mc_{d,\varepsilon}(3m)^{n}+\left[\sum_ {\ell=0}^{n-1}m^{n+1-\ell}c_{d,\varepsilon}(3m)^{\ell}\right]+\left[\sum_{ \ell=1}^{n}m^{n+1-\ell}c_{d,\varepsilon}(3m)^{\ell-1}\right]\\ &=m^{n+1}c_{d,\varepsilon}\left[1+3^{n}+\sum_{\ell=0}^{n-1}3^{ \ell}+\sum_{\ell=1}^{n}3^{\ell-1}\right]=m^{n+1}c_{d,\varepsilon}\left[1+ \sum_{\ell=0}^{n}3^{\ell}+\sum_{\ell=1}^{n}3^{\ell-1}\right]\\ &\leq cm^{n+1}\left[1+2\sum_{\ell=0}^{n}3^{\ell}\right]=cm^{n+1} \left[1+2\frac{3^{n+1}-1}{3-1}\right]=c_{d,\varepsilon}(3m)^{n+1}.\end{split} \tag{179}\]

This, (173), (171), the definition of \(L\) (see (157)), and (170) completes the induction step. The proof of Lemma 4.12 is thus completed. 

## 5. DNN approximations of PIDEs

In Theorem 5.1 below we combine the result of Lemmas 3.3, 3.2, and 2.1 to prove the existence of a DNN that approximates the solution to (190) and whose number of parameters depend only polynomially on the dimension \(d\) and the reciprocal of the prescribed accuracy \(\epsilon\).
**Theorem 5.1**.: _Consider the notations given in Subsection 1.4, assume Setting 1.1, let \(T\in(0,\infty)\), \(b,c\in[2,\infty)\) satisfy that_

\[16\Big{(}1+\left|c^{\frac{1}{2}}(4c^{\frac{1}{2}}+2c^{\frac{1}{2}}T^{-\frac{3}{ 2}})\right|^{\frac{1}{2}}\Big{)}\leq\frac{b}{4}, \tag{180}\]

_for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(v\in\mathbb{R}^{d}\) let \(\beta_{\varepsilon}^{d}\in C(\mathbb{R}^{d},\mathbb{R}^{d})\), \(\sigma_{\varepsilon}^{d}\in C(\mathbb{R}^{d},\mathbb{R}^{d\times d})\), \(\Phi_{\beta_{\varepsilon}^{d}},\Phi_{\sigma_{\varepsilon}^{d},v}\in\mathbf{N}\) satisfy that \(\beta_{\varepsilon}^{d}=\mathcal{R}(\Phi_{\beta_{\varepsilon}^{d}})\), \(\sigma_{\varepsilon}^{d}(\cdot)v=\mathcal{R}(\Phi_{\sigma_{\varepsilon}^{d},v})\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(\gamma_{\varepsilon}^{d}\colon\mathbb{R}^{2d}\to\mathbb{R}^{d}\), \(F_{\varepsilon}^{d}\colon\mathbb{R}^{d\times d}\), \(G^{d}\colon\mathbb{R}^{d}\to\mathbb{R}^{d}\) be measurable and satisfy for all \(y,z\in\mathbb{R}^{d}\) that \(\gamma_{\varepsilon}^{d}(y,z)=F_{\varepsilon}^{d}(y)G^{d}(z)\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(v\in\mathbb{R}^{d}\) let \(\Phi_{F_{\varepsilon}^{d},v}\in\mathbf{N}\) satisfy \(F_{\varepsilon}^{d}(\cdot)v=\mathcal{R}(\Phi_{F_{\varepsilon}^{d},v})\), assume for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1]\), \(v\in\mathbb{R}^{d}\) that \(\mathcal{D}(\Phi_{\sigma_{\varepsilon}^{d},v})=\mathcal{D}(\Phi_{\sigma_{ \varepsilon}^{d},0})\) and \(\mathcal{D}(\Phi_{F_{\varepsilon}^{d},v})=\mathcal{D}(\Phi_{F_{\varepsilon}^{d },0})\), for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(g_{\varepsilon}^{d}\in C(\mathbb{R}^{d},\mathbb{R})\), \(\Phi_{g_{\varepsilon}^{d}}\in\mathbf{N}\) satisfy that \(\mathcal{R}(\Phi_{g_{\varepsilon}^{d}})=g_{\varepsilon}^{d}\), for every \(d\in\mathbb{N}\) let \(\beta^{d}\in C(\mathbb{R}^{d},\mathbb{R}^{d})\), \(\sigma^{d}\in C(\mathbb{R}^{d},\mathbb{R}^{d\times d})\), \(g^{d}\in C(\mathbb{R}^{d},\mathbb{R})\), let \(f\in C(\mathbb{R},\mathbb{R})\), for every \(d\in\mathbb{N}\) let \(\nu^{d}\colon\mathcal{B}(\mathbb{R}^{d}\setminus\{0\})\to[0,\infty)\) be a Levy measure, assume that for all \(d\in\mathbb{N}\) there exists \(C_{d}\in(0,\infty)\) such that for all \(x,y,z\in\mathbb{R}^{d}\), \(t\in[0,T]\) we have that_

\[\left\|\gamma^{d}(x,z)\right\|\leq C_{d}\left(1\wedge\|z\|^{2}\right),\quad \left\|\gamma^{d}(x,z)-\gamma^{d}(y,z)\right\|^{2}\leq C_{d}\|x-y\|^{2}\left(1 \wedge\|z\|^{2}\right), \tag{181}\]

_assume that for all \(d\in\mathbb{N}\), \(t\in[0,T]\), \(x,z\in\mathbb{R}^{d}\) the Jacobian matrix \((D_{x}\gamma^{d})(x,z)\) exists, assume that for all \(d\in\mathbb{N}\) there exists \(\lambda_{d}\in(0,\infty)\) such that for all \(t\in[0,T]\), \(x,z\in\mathbb{R}^{d}\), \(\delta\in[0,1]\) we have that_

\[\lambda_{d}\leq\left|\det(I_{d}+\delta(D_{x}\gamma^{d})(x,z))\right|, \tag{182}\]

_where \(I_{d}\) denotes the \(d\times d\) identity matrix, assume for all \(d\in\mathbb{N}\), \(x,y\in\mathbb{R}^{d}\), \(w_{1},w_{2}\in\mathbb{R}\), \(\varepsilon\in(0,1)\) that_

\[\begin{split}&\left\|\beta_{\varepsilon}^{d}(x)-\beta_{ \varepsilon}^{d}(y)\right\|^{2}+\left\|\sigma_{\varepsilon}^{d}(x)-\sigma_{ \varepsilon}^{d}(y)\right\|^{2}_{\mathrm{F}}+\int_{\mathbb{R}^{d}\setminus\{0 \}}\left\|\gamma_{\varepsilon}^{d}(x,z)-\gamma_{\varepsilon}^{d}(y,z))\right\|^ {2}\nu^{d}(dz)\\ &\leq c\|x-y\|^{2},\end{split} \tag{183}\]

\[\begin{split}&\left|f(w_{1})-f(w_{2})\right|^{2}\leq c|w_{1}-w_{ 2}|^{2},\quad\left|g_{\varepsilon}^{d}(x)-g_{\varepsilon}^{d}(y)\right|^{2} \leq cd^{c}T^{-1}\|x-y\|^{2},\end{split} \tag{184}\]

\[\begin{split}&\left\|\beta_{\varepsilon}^{d}(x)-\beta^{d}(x) \right\|^{2}+\left\|\sigma_{\varepsilon}^{d}(x)-\sigma^{d}(x)\right\|^{2}_{ \mathrm{F}}\\ &\quad+\int_{\mathbb{R}^{d}\setminus\{0\}}\left\|\gamma_{ \varepsilon}^{d}(x,z)-\gamma^{d}(x,z)\right\|^{2}\nu^{d}(dz)+\left|g_{ \varepsilon}^{d}(x)-g^{d}(x)\right|^{2}\\ &\leq\varepsilon cd^{c}(d^{c}+\|x\|^{2}),\end{split} \tag{185}\]

\[\begin{split}&\left\|\!\left|\mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}}) \right|\!\right|\!\right|+\left|\!\left|\!\left|\mathcal{D}(\Phi_{\sigma_{ \varepsilon}^{d},0})\right|\!\right|\!\right|+\left|\!\left|\!\left|\mathcal{D}( \Phi_{F_{\varepsilon}^{d},0})\right|\!\right|\!\right|+\left|\!\left|\! \left|\mathcal{D}(\Phi_{g_{\varepsilon}^{d}})\right|\!\right|\!\right|\! \right|+\left|\!\left|\!\left|\mathcal{D}(\Phi_{g_{\varepsilon}^{d}})\right| \!\right|\!\right|\!\right|\leq\frac{bd^{c}\varepsilon^{-c}}{4},\end{split} \tag{187}\]

\[\dim(\mathcal{D}(\Phi_{\beta_{\ell}^{d}}))+\dim(\mathcal{D}(\Phi_{\sigma_{\ell}^{d},0}))+\dim(\mathcal{D}(\Phi_{F_{\varepsilon}^{d},0}))+\dim(\mathcal{D}(\Phi_{g _{\ell}^{d}}))\leq\frac{bd^{c}\varepsilon^{-c}}{4}, \tag{188}\]

_and for every \(d\in\mathbb{N}\) let \(\Gamma^{d}\colon\mathcal{B}(\mathbb{R}^{d})\to[0,1]\) be a probability measure satisfying_

\[\left(\int_{\mathbb{R}^{d}}\|x\|^{4}\,\Gamma^{d}(dx)\right)^{\frac{1}{2}}\leq cd ^{c}. \tag{189}\]

_Then_

1. _[label=()]_
2. _for every_ \(d\in\mathbb{N}\) _there exists a unique viscosity solution_ \(u^{d}\colon[0,T]\times\mathbb{R}^{d}\to\mathbb{R}\) _to the PIDE_ \[\begin{cases}&(\frac{\partial}{\partial u}u^{d})(t,x)+\left\langle\beta^{d}(x ),(\nabla_{x}u^{d})(t,x)\right\rangle\\ &+\frac{1}{2}\mathrm{trace}\big{(}\sigma^{d}(t,x)(\sigma^{d}(t,x))^{ \top}\mathrm{Hess}_{x}u^{d}(t,x)\big{)}+f(u^{d}(t,x))\\ &+\int_{\mathbb{R}^{d}}\left(u^{d}(x+\gamma^{d}(x,z))-u^{d}(t,x)- \left\langle(\nabla_{x}u^{d})(t,x),\gamma^{d}(x,z)\right\rangle\right)\nu^{d}( dz)=0,\\ &\quad\forall t\in[0,T),x\in\mathbb{R}^{d},\\ & u^{d}(T,x)=g^{d}(x),\,\forall\,x\in\mathbb{R}^{d},\end{cases}\]
_satisfying that_ \(\sup_{s\in[0,T],y\in\mathds{R}^{d}}|\frac{u^{d}(s,y)}{1+\|y\|}|<\infty\) _and_
2. _there exists_ \((C_{\delta})_{\delta\in(0,1)}\subseteq(0,\infty)\)_,_ \(\eta\in(0,\infty)\)_,_ \((\Psi_{d\varepsilon})_{d\in\mathbb{N},\,\epsilon\in(0,1)}\) _such that for all_ \(d\in\mathbb{N}\)_,_ \(\epsilon\in(0,1)\) _we have that_ \(\mathcal{R}(\Psi_{d,\epsilon})\in C(\mathds{R}^{d},\mathbb{R})\)_,_ \(\mathcal{P}(\Psi_{d,\epsilon})\leq C_{\delta}\eta d^{3c+12c^{2}+2c(6+\delta)} \epsilon^{-6c-6-\delta}\)_, and_ \[\left(\int_{\mathds{R}^{d}}\left|(\mathcal{R}(\Psi_{d,\epsilon}))(x)-u^{d}(t, x)\right|^{2}\Gamma^{d}(dx)\right)^{\frac{1}{2}}\leq\epsilon.\] (191)

Proof of Theorem 5.1.: First, (183)-(186) (with \(\varepsilon\to 0\)) show for all \(d\in\mathbb{N}\), \(x,y\in\mathds{R}^{d}\) that

\[\left\|\beta^{d}(x)-\beta^{d}(y)\right\|^{2}+\left\|\sigma^{d}(x)-\sigma^{d}( y)\right\|_{\rm F}^{2}+\int_{\mathds{R}^{d}\setminus\{0\}}\left\|\gamma^{d}(x,z)- \gamma^{d}(y,z)\right\|^{2}\nu^{d}(dz)\leq c\|x-y\|^{2}, \tag{192}\]

\[|f(w_{1})-f(w_{2})|^{2}\leq c|w_{1}-w_{2}|^{2},\quad\left|g^{d}(x)-g^{d}(y) \right|^{2}\leq cd^{c}T^{-1}\|x-y\|^{2}, \tag{193}\]

\[\left\|\beta^{d}(0)\right\|^{2}+\left\|\sigma^{d}(0)\right\|_{\rm F}^{2}+\int _{\mathds{R}^{d}\setminus\{0\}}\left\|\gamma^{d}(0,z)\right\|^{2}\nu^{d}(dz) +T^{3}|f(0)|^{2}+T|g^{d}(0)|^{2}\leq cd^{c}, \tag{194}\]

Next, [24, Corollary 3.13] (applied for all \(\varepsilon\in(0,1)\) with \(L\gets c^{\frac{1}{2}}\), \(q\gets 2\), \(\epsilon\leftarrow(\varepsilon/2)^{\frac{1}{2}}\) in the notation of [24, Corollary 3.13]), (185), and (180) show that there exist \(f_{\varepsilon}\in C(\mathds{R},\mathbb{R})\), \(\Phi_{f_{\varepsilon}}\in\mathbb{N}\) such that for all \(\varepsilon\in(0,1)\), \(w_{1},w_{2}\in\mathds{R}\) that

\[\mathcal{R}(\Phi_{f_{\varepsilon}})=f_{\varepsilon},\quad|f_{ \varepsilon}(w_{1})-f_{\varepsilon}(w_{1})|^{2}\leq c|w_{1}-w_{2}|^{2},\quad \dim(\mathcal{D}(\Phi_{f_{\varepsilon}}))=3\leq\frac{bd^{c}\varepsilon^{-c}} {4}, \tag{195}\]

\[\left\|\!\left|\mathcal{D}(\Phi_{f_{\varepsilon}})\right|\!\right| \leq 16\Big{(}1+\left|c^{\frac{1}{2}}(4c^{\frac{1}{2}}+2|f(0)|) \right|^{\frac{1}{2}}\Big{)}\varepsilon^{-2} \tag{196}\] \[\leq 16\Big{(}1+\left|c^{\frac{1}{2}}(4c^{\frac{1}{2}}+2c^{\frac{ 1}{2}}T^{-\frac{3}{2}})\right|^{\frac{1}{2}}\Big{)}\varepsilon^{-2}\leq\frac{ b\varepsilon^{-2}}{4},\]

and

\[|f(w_{1})-f_{\varepsilon}(w_{1})|^{2}\leq\frac{\varepsilon^{2}}{2}(1+|w_{1}| ^{2})^{2}\leq\varepsilon(1+|w_{1}|^{4}). \tag{197}\]

This, (187), and (188) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) that

\[2d+\left|\!\left|\!\left|\mathcal{D}(\Phi_{\varrho_{\varepsilon}^{d}}) \right|\!\right|\!\right|+\left|\!\left|\!\left|\mathcal{D}(\Phi_{\varrho_{ \varepsilon}^{d},0})\right|\!\right|\!\right|+\left|\!\left|\!\left|\mathcal{D}( \Phi_{F_{\varepsilon}^{d},0})\right|\!\right|\!\right|+\left|\!\left|\!\left| \mathcal{D}(\Phi_{g_{\varepsilon}^{d}})\right|\!\right|\!\right|+\left|\! \left|\!\left|\mathcal{D}(\Phi_{f_{\varepsilon}})\right|\!\right|\!\right|\leq bd ^{c}\varepsilon^{-c} \tag{198}\]

and

\[\dim(\mathcal{D}(\Phi_{\varrho_{\varepsilon}^{d}}))+\dim(\mathcal{D}(\Phi_{ \sigma_{\varepsilon}^{d},0}))+\dim(\mathcal{D}(\Phi_{F_{\varepsilon}^{d},0})) +\dim(\mathcal{D}(\Phi_{g_{\varepsilon}^{d}}))+\dim(\mathcal{D}(\Phi_{f_{ \varepsilon}}))\leq bd^{c}\varepsilon^{-c}. \tag{199}\]

Furthermore, (197), (185), and the triangle inequality show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) that \(|f_{\varepsilon}(0)|\leq\sqrt{\varepsilon}+|f(0)|\leq 1+|f(0)|\) and hence

\[\left\|\beta_{\varepsilon}^{d}(0)\right\|^{2}+\left\|\sigma_{\varepsilon}^{d}(0 )\right\|_{\rm F}^{2}+\int_{\mathds{R}^{d}\setminus\{0\}}\left\|\gamma_{ \varepsilon}^{d}(0,z)\right\|^{2}\nu^{d}(dz)+T^{3}|f_{\varepsilon}(0)|+T|g_{ \varepsilon}^{d}(0)|^{2}\leq cd^{c}. \tag{200}\]

Next, for every \(K\in\mathbb{N}\) let \(\lfloor\cdot\rfloor_{K}\colon\mathbb{R}\to\mathbb{R}\) satisfy for all \(t\in\mathbb{R}\) that \(\lfloor t\rfloor_{K}=\max(\{0,\frac{T}{K},\frac{2T}{K},\ldots,T\}\cap((-\infty,t )\cup\{0\}))\), let \((\Omega,\mathcal{F},\mathbb{P},(\mathbb{F}_{t})_{t\in[0,T]})\) be a probability space satisfying the usual conditions, let \(\Theta=\cup_{n\in\mathbb{N}}\mathbb{Z}^{n}\), for every \(d\in\mathbb{N}\) let \(W^{d,\theta}\colon\Omega\times[0,T]\to\mathbb{R}^{d}\), \(\theta\in\Theta\), be identically independently distributed standard \((\mathbb{F}_{t})_{t\in[0,T]}\)-Brownian motions, for every \(d\in\mathbb{N}\) let \(N^{d,\theta}\), \(\theta\in\Theta\), be independent \((\mathbb{F}_{t})_{t\in[0,T]}\)-Poisson random measures on \([0,\infty)\times(\mathds{R}^{d}\setminus\{0\})\) with intensity \(\nu^{d}\), for every \(d\in\mathbb{N}\), \(\theta\in\Theta\) let \(\tilde{N}^{d,\theta}(dt,dz)=N^{d,\theta}(dt,dz)-dt\,\nu^{d}(dz)\), assume for all \(d\in\mathbb{N}\) that \(\mathcal{F}_{0}\), \((N^{d,\theta})_{\theta\in\Theta}\) and \((W^{d,\theta})_{\theta\in\Theta}\), are independent, for every \(d,K\in\mathbb{N}\), \(\theta\in\Theta\), \(x\in\mathds{R}^{d}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\) let \((X_{s}^{d,\theta,K,\varepsilon,t,x})_{s\in[t,T]}\) satisfy that

\[X_{s}^{d,\theta,K,\varepsilon,t,x} =x+\int_{t}^{s}\beta_{\varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}}^{d,\theta,K,\varepsilon,t,x})\,dr+\int_{t}^{s}\sigma_{\varepsilon}^{d}(X_{\max\{t,[ r-]_{K}\}}^{d,\theta,K,\varepsilon,t,x})\,dW_{r}^{d,\theta} \tag{201}\] \[\quad+\int_{t}^{s}\int_{\mathds{R}^{d}\setminus\{0\}}^{\prime} \sigma_{\varepsilon}^{d}(X_{\max\{t,[r-]_{K}\}}^{d,\theta,K,\varepsilon,t,x})\, \tilde{N}^{d,\theta}(dr,dz),\]

let \(\theta\colon\Omega\to[0,1]\), \(\theta\in\Theta\), be i.i.d random variables which satisfy for all \(t\in(0,1)\) that \(\mathbb{P}\left(\mathfrak{t}^{0}\leq t\right)=t\), for every \(\theta\in\Theta,t\in[0,T]\) let \(\mathfrak{T}_{t}^{d}\colon\Omega\to\mathbb{R}\) satisfy for all \(\
\((N^{d,\theta})_{\theta\in\Theta}\) and \((W^{d,\theta})_{\theta\in\Theta}\) are independent, for every \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\) let \(U_{n,m}^{d,\theta,K,\varepsilon}\colon[0,T]\times\mathbb{R}^{d}\times\Omega \to\mathbb{R}\), \(\theta\in\Theta\), \(n,m\in\mathbb{Z}\), satisfy for all \(\theta\in\Theta\), \(n\in\mathbb{N}_{0}\), \(m\in\mathbb{N}\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) that

\[\begin{split}& U_{n,m}^{d,\theta,K,\varepsilon}(t,x)=\frac{ \mathbbm{1}_{\mathbb{N}}(n)}{m^{n}}\sum_{i=1}^{m^{n}}g_{\varepsilon}^{d}\Big{(} X_{T}^{d,(\theta,0,-i),K,\varepsilon,t,x}\Big{)}\\ &\quad+\sum_{\ell=0}^{n-1}\frac{(T-t)}{m^{n-\ell}}\sum_{i=1}^{m^ {n-\ell}}\Big{(}f_{\varepsilon}\circ U_{\ell,m}^{d,(\theta,\ell,i),K,\varepsilon }-\mathbbm{1}_{\mathbb{N}}(\ell)f_{\varepsilon}\circ U_{\ell-1,m}^{d,(\theta,-\ell,i),K,\varepsilon}\Big{)}\Big{(}\mathfrak{z}_{t}^{(\theta,\ell,i)},X_{ \mathfrak{z}_{t}^{(\theta,\ell,i)}}^{d,(\theta,\ell,i),K,\varepsilon,t,x} \Big{)}.\end{split} \tag{202}\]

A standard result on existence and uniqueness of SDEs with jumps (cf., e.g., [29, Theorem 9.1]), (183), and (185) show that for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) there exist adapted cadlag processes \((X_{s}^{d,\varepsilon,t,x})_{s\in[t,T]},(X_{s}^{d,t,x})_{s\in[t,T]}\) such that for all \(s\in[t,T]\) we have \(\mathbb{P}\)-a.s. that

\[X_{s}^{d,t,x}=x+\int_{t}^{s}\beta^{d}(X_{r-}^{d,t,x})dr+\int_{t}^{s}\sigma^{d} (X_{r-}^{d,t,x})dW_{r}^{d,0}+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\}} \gamma^{d}(X_{r-}^{d,t,x},z)\tilde{N}^{d,0}(dr,dz) \tag{203}\]

and

\[X_{s}^{d,\varepsilon,t,x}=x+\int_{t}^{s}\beta_{\varepsilon}^{d}(X_{r-}^{d, \varepsilon,t,x})dr+\int_{t}^{s}\sigma_{\varepsilon}^{d}(X_{r-}^{d, \varepsilon,t,x})dW_{r}^{d,0}+\int_{t}^{s}\int_{\mathbb{R}^{d}\setminus\{0\} }\gamma_{\varepsilon}^{d}(X_{r-}^{d,\varepsilon,t,x},z)\tilde{N}^{d,0}(dr,dz). \tag{204}\]

Observe that for all \(d,K\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) we have that

\[\max\left\{\mathbb{E}\bigg{[}d^{c}+\left\|X_{s}^{d,K,\varepsilon,t,x}\right\|^ {2}\bigg{]},\mathbb{E}\bigg{[}d^{c}+\left\|X_{s}^{d,\varepsilon,t,x}\right\|^ {2}\bigg{]},\mathbb{E}\bigg{[}d^{c}+\left\|X_{s}^{d,t,x}\right\|^{2}\bigg{]} \right\}\leq(d^{c}+\|x\|^{2})e^{7\varepsilon(s-t)} \tag{205}\]

(cf. Lemmas 2.1 and 3.2). Next, (185) and (184) show for all \(d\in\mathbb{N}\), \(\varepsilon\in(0,1)\), \(x\in\mathbb{R}^{d}\) that

\[\left|g_{\varepsilon}^{d}(x)\right|\leq\left|g_{\varepsilon}^{d}(0)\right|+(cd ^{c}T^{-1})^{\frac{1}{2}}\|x\|\leq(cd^{c}T^{-1})^{\frac{1}{2}}+(cd^{c}T^{-1})^{ \frac{1}{2}}\|x\|\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}. \tag{206}\]

This and (186) (with \(\varepsilon\to 0\)) show for all \(d\in\mathbb{N}\), \(x\in\mathbb{R}^{d}\) that

\[\left|g^{d}(x)\right|\leq 2(cd^{c}T^{-1})^{\frac{1}{2}}(d^{c}+\|x\|^{2})^{\frac{1}{2}}. \tag{207}\]

This, (206), (205), (184), (195), and [23, Proposition 2.2] (applied with \(V\leftarrow([0,T]\times\mathbb{R}^{d}\ni(t,x)\mapsto e^{4.5c(T-t)}(d^{c}+\|x\|^ {2})^{\frac{1}{2}}\in(0,\infty)\)) in the notation of [23, Proposition 2.2]) show that for all \(\varepsilon\in(0,1)\), \(d,K\in\mathbb{N}\) there exist measurable functions \(u^{d,K,\varepsilon},u^{d,\varepsilon},u^{d}\colon[0,T]\times\mathbb{R}^{d}\) such that for all \(t\in[0,T]\), \(x\in\mathbb{R}^{d}\) we have that \(\mathbb{E}\left[\left|g_{\varepsilon}^{d}(X_{T}^{d,K,\varepsilon,t,x}) \right|\right]+\int_{t}^{T}\mathbb{E}\Big{[}\big{|}f_{\varepsilon}(u^{d,K, \varepsilon}(r,X_{r}^{d,K,\varepsilon,t,x}))\big{|}\Big{]}\,dr+\mathbb{E} \Big{[}\big{|}g_{\varepsilon}^{d}(X_{T}^{d,\varepsilon,t,x})\big{|}\Big{]}+\int_ {t}^{T}\mathbb{E}\Big{[}\big{|}f_{\varepsilon}(u^{d,\varepsilon}(r,X_{r}^{d, \varepsilon,t,x}))\big{|}\Big{]}+\mathbb{E}\Big{[}\big{|}g_{\varepsilon}^{d}(X_{T }^{d,\varepsilon})\big{|}\Big{]}+\int_{t}^{T}\mathbb{E}\Big{[}\big{|}f(u^{d }(r,X_{r}^{d,t,x}))\big{|}\Big{]}<\infty\),

\[\sup_{s\in[0,T]}\sup_{y\in\mathbb{R}^{d}}\frac{|u^{d,K,\varepsilon}(s,y)|+|u^{ d,\varepsilon}(s,y)|+|u^{d}(s,y)|}{1+\|y\|}<\infty, \tag{208}\]

\[u^{d,K,\varepsilon}(t,x)=\mathbb{E}\Big{[}g_{\varepsilon}^{d}(X_{T}^{d,K, \varepsilon,t,x})\Big{]}+\int_{t}^{T}\mathbb{E}\Big{[}f_{\varepsilon}(u^{d,K, \varepsilon}(r,X_{r}^{d,K,\varepsilon,t,x}))\Big{]}\,dr, \tag{209}\]

\[u^{d,\varepsilon}(t,x)=\mathbb{E}\Big{[}g_{\varepsilon}^{d}(X_{T}^{d, \varepsilon,t,x})\Big{]}+\int_{t}^{T}\mathbb{E}\Big{[}f_{\varepsilon}(u^{d, \varepsilon}(r,X_{r}^{d,\varepsilon,t,x}))\Big{]}\,dr, \tag{210}\]

and

\[u^{d}(t,x)=\mathbb{E}\Big{[}g^{d}(X_{T}^{d,t,x})\Big{]}+\int_{t}^{T}\mathbb{E} \Big{[}f(u^{d}(r,X_{r}^{d,t,x}))\Big{]}\,dr. \tag{211}\]

This, a result on existence and uniqueness of viscosity solutions to PIDEs (see [31, Propositions 5.4 and 5.16]), and the assumptions of Theorem 5.1 show (i).

Next, let \(c_{1},c_{2}\in\mathbb{R}\), \((\varepsilon_{d,\varepsilon})_{d\in\mathbb{N},\varepsilon\in(0,1)}\subseteq \mathbb{R}\), \((N_{d,\varepsilon})_{d\in\mathbb{N},\varepsilon\in(0,1)}\subseteq\mathbb{N}\), \((C_{\delta})_{\delta\in(0,1)}\subseteq[0,\infty]\) satisfy for all \(d\in\mathbb{N}\), \(\delta,\epsilon\in(0,1)\) that

\[c_{1}=6(cT^{-1})^{\frac{1}{2}}+12c^{\frac{2}{2}}(T+2)e^{21cT+5cT^{2}}T^{\frac{1}{2} }+2ce^{24cT+5cT^{2}},\quad c_{2}=2cc_{1}, \tag{212}\]
\[c_{2}d^{2c}|\varepsilon_{d,\epsilon}|^{\frac{1}{2}}=\frac{\epsilon}{2},\quad N_{ d,\epsilon}=\min\left\{n\in\mathbb{N}\cap[2,\infty)\colon c_{2}d^{2c}\left(\frac{e^{12cTn+ \frac{n}{2}}}{n^{\frac{n}{2}}}+\frac{1}{n^{\frac{n}{2}}}\right)\leq\frac{ \epsilon}{2}\right\}, \tag{213}\]

and

\[C_{\delta}=\sup_{n\in[2,\infty)}\left[\left(\frac{e^{(12cT+0.5)(n-1)}}{(n-1)^ {\frac{n-1}{2}}}\right)^{6+\delta}(3n)^{3n+1}\right]. \tag{214}\]

Then the triangle inequality, Lemmas 3.3, 3.2, and 2.1 show for all \(d,K\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(n,m\in\mathbb{N}\) that

\[\left(\mathbb{E}\bigg{[}\Big{|}U_{n,m}^{d,\theta,K,\varepsilon}(t,x)-u^{d}(t,x)\Big{|}^{2}\bigg{]}\right)^{\frac{1}{2}} \tag{215}\] \[\leq 6e^{\frac{n}{2}}m^{-\frac{n}{2}}e^{12cTn}(cd^{c}T^{-1})^{\frac{1 }{2}}\left(d^{c}+\|x\|^{2}\right)^{\frac{1}{2}}\] \[+12c^{\frac{3}{2}}d^{\frac{5}{2}}(T+2)e^{21cT+5cT^{2}}(d^{c}+\|x \|^{2})^{\frac{1}{2}}\frac{T^{\frac{1}{2}}}{K^{\frac{1}{2}}}+2cd^{c}\varepsilon ^{\frac{1}{2}}(d^{c}+\|x\|^{2})e^{24cT+5cT^{2}}\] \[\leq \left[6(cd^{c}T^{-1})^{\frac{1}{2}}+12c^{\frac{3}{2}}d^{\frac{5} {2}}(T+2)e^{21cT+5cT^{2}}T^{\frac{1}{2}}+2cd^{c}e^{24cT+5cT^{2}}\right](d^{c} +\|x\|^{2})\left(\varepsilon^{\frac{1}{2}}+\frac{e^{12cTn+\frac{m}{2}}}{m^{ \frac{n}{2}}}+\frac{1}{K^{\frac{1}{2}}}\right)\] \[\leq \left[6(cT^{-1})^{\frac{1}{2}}+12c^{\frac{3}{2}}(T+2)e^{21cT+5cT^ {2}}T^{\frac{1}{2}}+2ce^{24cT+5cT^{2}}\right]d^{c}(d^{c}+\|x\|^{2})\left( \varepsilon^{\frac{1}{2}}+\frac{e^{12cTn+\frac{m}{2}}}{m^{\frac{n}{2}}}+ \frac{1}{K^{\frac{1}{2}}}\right)\] \[\leq c_{1}d^{c}\left(\varepsilon^{\frac{1}{2}}+\frac{e^{12cTn+ \frac{m}{2}}}{m^{\frac{n}{2}}}+\frac{1}{K^{\frac{1}{2}}}\right)(d^{c}+\|x\|^{2 }).\]

This, the triangle inequality, (189), and (212) show for all \(d,K\in\mathbb{N}\), \(\theta\in\Theta\), \(\varepsilon\in(0,1)\), \(n,m\in\mathbb{N}\) that

\[\left(\int_{\mathbb{R}^{d}}\mathbb{E}\left[\left|U_{n,m}^{d,\theta,K,\varepsilon}(t,x)-u^{d}(t,x)\right|^{2}\right]\Gamma^{d}(dx)\right)^{\frac{ 1}{2}} \tag{216}\] \[\leq c_{1}d^{c}\left(\varepsilon^{\frac{1}{2}}+\frac{e^{12cTn+ \frac{m}{2}}}{m^{\frac{n}{2}}}+\frac{1}{K^{\frac{1}{2}}}\right)\left(\int_{ \mathbb{R}^{d}}(d^{c}+\|x\|^{2})^{2}\Gamma^{d}(dx)\right)^{\frac{1}{2}}\] \[\leq c_{1}d^{c}\left(\varepsilon^{\frac{1}{2}}+\frac{e^{12cTn+ \frac{m}{2}}}{m^{\frac{n}{2}}}+\frac{1}{K^{\frac{1}{2}}}\right)\left(d^{c}+ \left(\int_{\mathbb{R}^{d}}\|x\|^{4}\,\Gamma^{d}(dx)\right)^{\frac{1}{2}}\right)\] \[\leq c_{1}d^{c}\left(\varepsilon^{\frac{1}{2}}+\frac{e^{12cTn+ \frac{m}{2}}}{m^{\frac{n}{2}}}+\frac{1}{K^{\frac{1}{2}}}\right)2cd^{c}=c_{2}d^ {2c}\left(\varepsilon^{\frac{1}{2}}+\frac{e^{12cTn+\frac{m}{2}}}{m^{\frac{n}{2 }}}+\frac{1}{K^{\frac{1}{2}}}\right).\]

This, Fubini's theorem, and (213) show for all \(d\in\mathbb{N}\), \(\epsilon\in(0,1)\) that

\[\left(\mathbb{E}\left[\int_{\mathbb{R}^{d}}\left|U_{n,n}^{d,\theta,n,\varepsilon}(t,x)-u^{d}(t,x)\right|^{2}\Gamma^{d}(dx)\right]\right)^{\frac{ 1}{2}}\Big{|}_{\begin{subarray}{c}n=N_{d,\epsilon}\\ \varepsilon=d_{\epsilon}\end{subarray}} \tag{217}\] \[=\left(\int_{\mathbb{R}^{d}}\mathbb{E}\left[\left|U_{n,n}^{d, \theta,n,\varepsilon}(t,x)-u^{d}(t,x)\right|^{2}\right]\Gamma^{d}(dx)\right)^{ \frac{1}{2}}\Big{|}_{\begin{subarray}{c}n=N_{d,\epsilon}\\ \varepsilon=d_{\epsilon}\end{subarray}}\] \[\leq c_{2}d^{2c}\left(\varepsilon^{\frac{1}{2}}+\frac{e^{12cTn+ \frac{m}{2}}}{n^{\frac{n}{2}}}+\frac{1}{n^{\frac{n}{2}}}\right)\Big{|}_{ \begin{subarray}{c}n=N_{d,\epsilon}\\ \varepsilon=d_{\epsilon}\end{subarray}}\] \[\leq\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon.\]

Then for all \(d\in\mathbb{N}\), \(\epsilon\in(0,1)\) there exists \(\omega_{d,\epsilon}\in\Omega\) such that

\[\int_{\mathbb{R}^{d}}\left|U_{n,n}^{d,\theta,n^{n},\varepsilon}(t,x,\omega_{d, \epsilon})-u^{d}(t,x)\right|^{2}\Gamma^{d}(dx)\Big{|}_{\begin{subarray}{c}n=N_{ d,\epsilon}\\ \varepsilon=d_{\epsilon}\end{subarray}}\leq\epsilon^{2}. \tag{218}\]
Next, Lemma 4.12, (199), and (198) show for all \(d\in\mathbb{N}\), \(\epsilon\in(0,1)\) that there exists \(\Psi_{d,\epsilon}\in\mathbb{N}\) such that

1. we have for all \(t\in[0,T]\), \(\theta\in\Theta\) that \[\begin{split}\dim(\mathcal{D}(\Psi_{d,\epsilon}))& =(n+1)\left[n^{n}\left(\max\left\{\dim(\mathcal{D}(\Phi_{g_{ \epsilon}^{d}})),\dim(\mathcal{D}(\Phi_{g_{\epsilon}^{d}}))\right\}-1 \right)+1\right]\\ &\quad+n(\dim(\mathcal{D}(\Phi_{f_{\epsilon}}))-2)+\dim(\mathcal{ D}(\Phi_{g_{\epsilon}^{d}}))-1\Big{|}_{n=N_{d,\epsilon},\varepsilon=\varepsilon_{d, \epsilon}}\\ &\leq 2nn^{n}bd^{c}\varepsilon^{-c}+nbd^{c}\varepsilon^{-c}+bd^{c} \varepsilon^{-c}\big{|}_{n=N_{d,\epsilon}}\leq 4n^{n}nbd^{c}\varepsilon^{-c} \big{|}_{n=N_{d,\epsilon},\varepsilon=\varepsilon_{d,\epsilon}},\end{split}\] (219)
2. we have for all \(t\in[0,T]\), \(\theta\in\Theta\) that \[\begin{split}\left|\!\left|\!\left|\mathcal{D}(\Psi_{d, \epsilon})\right|\!\right|\!\right|&\leq\left(2d+\left|\!\left| \!\left|\!\left|\mathcal{D}(\Phi_{g_{\epsilon}^{d}})\right|\!\right|\! \right|\!\right|+\left|\!\left|\!\left|\mathcal{D}(\Phi_{g_{\epsilon}^{d},0}) \right|\!\right|\!\right|\\ &\quad\quad+\left|\!\left|\!\left|\mathcal{D}(\Phi_{F_{d}^{d},0} )\right|\!\right|\!\right|+\left|\!\left|\!\left|\mathcal{D}(\Phi_{g_{\epsilon }^{d}})\right|\!\right|\!\right|+\left|\!\left|\!\left|\mathcal{D}(\Phi_{f_{ \epsilon}})\right|\!\right|\!\right|+\left|\!\left|\!\left|\mathcal{D}(\Phi_{f _{\epsilon}})\right|\!\right|\!\right)(3n)^{n}\big{|}_{n=N_{d,\epsilon}, \varepsilon=\varepsilon_{d,\epsilon}}\\ &\leq bd^{c}\varepsilon^{-c}(3n)^{n}\big{|}_{n=N_{d,\epsilon}, \varepsilon=\varepsilon_{d,\epsilon}},\end{split}\] (220) and
3. we have for all \(t\in[0,T]\), \(\theta\in\Theta\), \(x\in\mathbb{R}^{d}\) that \(U_{n,n}^{d,\theta,n^{n},\varepsilon}(t,x,\omega_{d,\epsilon})\Big{|}_{ \begin{subarray}{c}n=N_{d,\epsilon}\\ \varepsilon=\varepsilon_{d,\epsilon}\end{subarray}}=(\mathcal{R}(\Psi_{d, \epsilon}))(x)\).

This and (218) show for all \(d\in\mathbb{N}\), \(\epsilon\in(0,1)\) that

\[\begin{split}&\int_{\mathbb{R}^{d}}\left|(\mathcal{R}(\Psi_{d, \epsilon}))(x)-u^{d}(t,x)\right|^{2}\Gamma^{d}(dx)\leq\epsilon^{2}.\end{split} \tag{221}\]

Furthermore, the fact that \(\forall\,\Phi\in\mathbb{N}\colon\mathcal{P}(\Phi)\leq 2\dim(\mathcal{D}(\Phi)) \!\left|\!\left|\!\left|\mathcal{D}(\Phi)\right|\!\right|\!\right|^{2}\), (219), and (220) show for all \(d\in\mathbb{N}\), \(\epsilon\in(0,1)\) that

\[\begin{split}&\mathcal{P}(\Psi_{d,\epsilon})\leq 2\dim(\mathcal{ D}(\Psi_{d,\epsilon}))\!\left|\!\left|\mathcal{D}(\Psi_{d,\epsilon})\right|\! \right|\!\right|^{2}\leq 2\cdot 4n^{n}nbd^{c}\varepsilon^{-c}\left(bd^{c} \varepsilon^{-c}(3n)^{n}\right)^{2}\!\big{|}_{n=N_{d,\epsilon}}\\ &=8n^{n}nb^{2}d^{3c}|\varepsilon_{d,\epsilon}|^{-3c}(3n)^{2n} \big{|}_{n=N_{d,\epsilon},\varepsilon=\varepsilon_{d,\epsilon}}\end{split} \tag{222}\]

Recall that in (213) we have for all \(d\in\mathbb{N}\), \(\epsilon\in(0,1)\) that \(c_{2}d^{2c}|\varepsilon_{d,\epsilon}|^{\frac{1}{2}}=\frac{\epsilon}{2}\). Hence, for all \(d\in\mathbb{N}\), \(\epsilon\in(0,1)\) we have that \(\varepsilon_{d,\epsilon}=\frac{\epsilon^{2}}{4}|c_{2}|^{-2}d^{-4c}\). This, (222), and (214) show for all \(d\in\mathbb{N}\), \(\epsilon,\delta\in(0,1)\) that

\[\begin{split}&\mathcal{P}(\Psi_{d,\epsilon})\leq 8n^{n}nb^{2}d^{3c}| \varepsilon_{d,\epsilon}|^{-3c}(3n)^{2n}\big{|}_{n=N_{d,\epsilon}}\\ &\leq 8b^{2}d^{3c}|\varepsilon_{d,\epsilon}|^{-3c}(3n)^{3n+1} \Big{|}_{n=N_{d,\epsilon}}\\ &\leq 8b^{2}d^{3c}\left[\frac{\epsilon^{2}}{4}|c_{2}|^{-2}d^{-4c} \right]^{-3c}(3n)^{3n+1}\Big{|}_{n=N_{d,\epsilon}}\\ &\leq 4^{3c+2}b^{2}d^{3c+12c^{2}}|c_{2}|^{6c}\epsilon^{-6c}(3N_{d, \epsilon})^{3N_{d,\epsilon}+1}\\ &\leq 4^{3c+2}b^{2}d^{3c+12c^{2}}|c_{2}|^{6c}\epsilon^{-6c-6-\delta }\epsilon^{6+\delta}(3N_{d,\epsilon})^{3N_{d,\epsilon}+1}\\ &\leq 4^{3c+2}b^{2}d^{3c+12c^{2}}|c_{2}|^{6c}\epsilon^{-6c-6-\delta }\left(\frac{4c_{2}d^{2c}e^{(12cT+0.5)(N_{d,\epsilon}-1)}}{(N_{d,\epsilon}-1) \frac{N_{d,\epsilon}-1}{2}}\right)^{6+\delta}(3N_{d,\epsilon})^{3N_{d, \epsilon}+1}\\ &\leq 4^{3c+8+\delta}b^{2}d^{3c+12c^{2}+2c(6+\delta)}|c_{2}|^{6c+2c(6+ \delta)}\epsilon^{-6c-6-\delta}C_{\delta}.\end{split} \tag{223}\]

This, (221), the fact that \(c_{2}\) does not depend on \(d\) (see (212)), and the fact that \(\forall\,\delta\in(0,1)\colon C_{\delta}<\infty\) (cf. (171) in [8]) complete the proof of Theorem 5.1. 

Proof of Theorem 1.2.: Let \(b,\tilde{c}\in[2,\infty)\) satisfy that

\[c\leq\tilde{c},\quad Ted^{c}\leq\tilde{c}d^{\tilde{c}},\quad T^{3}(c^{\frac{1}{2}} d^{\frac{5}{2}}+1)\leq\tilde{c}d^{\tilde{c}}, \tag{224}\]
and

\[16\Big{(}1+\Big{|}(3\tilde{c})^{\frac{1}{2}}(4(3\tilde{c})^{\frac{1}{2}}+2(3\tilde{ c})^{\frac{1}{2}}T^{-\frac{3}{2}})\Big{|}^{\frac{1}{2}}\Big{)}\leq\frac{b}{4}. \tag{225}\]

Then Theorem 1.2 follows from Theorem 5.1 (applied with \(c\gets 3\tilde{c}\), \((\Gamma^{d})_{d\in\mathbb{N}}\leftarrow(\int_{(\cdot)\cap[0,1]^{d}}dx)_{d\in \mathbb{N}}\) in the notation of Theorem 5.1). This can be easily checked as follows. From (183)-(188), (224), and (225) it follows that for all \(d\in\mathbb{N}\), \(x,y\in\mathbb{R}^{d}\), \(w_{1},w_{2}\in\mathbb{R}\), \(\varepsilon\in(0,1)\) we have that

\[\Big{\|}\beta_{\varepsilon}^{d}(x)-\beta_{\varepsilon}^{d}(y)\Big{\|}^{2}+ \Big{\|}\sigma_{\varepsilon}^{d}(x)-\sigma_{\varepsilon}^{d}(y)\Big{\|}_{ \mathbb{F}}^{2}+\int_{\mathbb{R}^{d}\setminus\{0\}}\Big{\|}\gamma_{ \varepsilon}^{d}(x,z)-\gamma_{\varepsilon}^{d}(y,z))\Big{\|}^{2}\,\nu^{d}(dz) \leq\tilde{c}\|x-y\|^{2}, \tag{226}\]

\[|f(w_{1})-f(w_{2})|^{2}\leq\tilde{c}|w_{1}-w_{2}|^{2},\quad\Big{|}g_{ \varepsilon}^{d}(x)-g_{\varepsilon}^{d}(y)\Big{|}^{2}\leq cd^{\tilde{c}}\|x- y\|^{2}\leq\tilde{c}d^{\tilde{c}}T^{-1}\|x-y\|^{2}, \tag{227}\]

\[\Big{\|}\beta_{\varepsilon}^{d}(0)\Big{\|}^{2}+\Big{\|}\sigma_{\varepsilon}^ {d}(0)\Big{\|}_{\mathbb{F}}^{2}+\int_{\mathbb{R}^{d}\setminus\{0\}}\Big{\|} \gamma_{\varepsilon}^{d}(0,z)\Big{\|}^{2}\,\nu^{d}(dz)\leq\tilde{c}d^{\tilde{ c}}, \tag{228}\]

\[T^{3}(|f(0)|+1)^{2}\leq T^{3}(c^{\frac{1}{2}}d^{\tilde{c}}+1)\leq\tilde{c}d^ {\tilde{c}}, \tag{229}\]

\[T|g_{\varepsilon}^{d}(0)|^{2}\leq Tcd^{c}\leq\tilde{c}d^{\tilde{c}}, \tag{230}\]

\[\Big{\|}\beta_{\varepsilon}^{d}(x)-\beta^{d}(x)\Big{\|}^{2}+\Big{\|}\sigma_{ \varepsilon}^{d}(x)-\sigma^{d}(x)\Big{\|}_{\mathbb{F}}^{2}+\int_{\mathbb{R}^ {d}\setminus\{0\}}\Big{\|}\gamma_{\varepsilon}^{d}(x,z)-\gamma^{d}(x,z) \Big{\|}^{2}\,\,\nu^{d}(dz)+\Big{|}g_{\varepsilon}^{d}(x)-g^{d}(x)\Big{|}^{2}\]

\[\leq\varepsilon\tilde{c}d^{\tilde{c}}(d^{\tilde{c}}+\|x\|^{2}), \tag{231}\]

\[\Big{|}\Big{|}\Big{|}\mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}})\Big{|}\Big{|} +\Big{|}\Big{|}\mathcal{D}(\Phi_{\sigma_{\varepsilon}^{d},0})\Big{|}\Big{|}+ \Big{|}\Big{|}\mathcal{D}(\Phi_{F_{\varepsilon}^{d},0})\Big{|}\Big{|}+\Big{|} \Big{|}\mathcal{D}(\Phi_{g_{\varepsilon}^{d}})\Big{|}\Big{|} \tag{232}\]

and

\[\begin{array}{l}\dim(\mathcal{D}(\Phi_{\beta_{\varepsilon}^{d}}))+\dim( \mathcal{D}(\Phi_{\sigma_{\varepsilon}^{d},0}))+\dim(\mathcal{D}(\Phi_{F_{ \varepsilon}^{d},0}))+\dim(\mathcal{D}(\Phi_{g_{\varepsilon}^{d}}))\\ \leq\mathcal{P}(\Phi_{\beta_{\varepsilon}^{d}})+\mathcal{P}(\Phi_{\sigma_{ \varepsilon}^{d},0})+\mathcal{P}(\Phi_{F_{\varepsilon}^{d},0})+\mathcal{P}( \Phi_{g_{\varepsilon}^{d}})\leq\frac{bd^{c}\varepsilon^{-c}}{4}.\end{array} \tag{233}\]

In addition, for every \(d\in\mathbb{N}\) let \(\Gamma^{d}\colon\mathcal{B}(\mathbb{R}^{d})\to[0,1]\) be a probability measure satisfying \(\Gamma^{d}(\cdot)=\int_{(\cdot)\cap[0,1]^{d}}dx\). Then for all \(d\in\mathbb{N}\) we have that

\[\left(\int_{\mathbb{R}^{d}}\|x\|^{4}\,\Gamma^{d}(dx)\right)^{\frac{1}{2}}= \left(\int_{[0,1]^{d}}\|x\|^{4}\,dx\right)^{\frac{1}{2}}\leq d^{2}\leq\tilde{c }d^{\tilde{c}}. \tag{234}\]

The proof of Theorem 1.2 is thus completed. 
