Ergodic and strong Feller properties

of affine processes111 Supported by the National Natural Science Foundation of
China (No. 11531001).

Shukai Chen and Zenghu Li

Department of Mathematics, Beijing Normal University,

Beijing 100875, People’s Republic of China

E-mail<EMAIL_ADDRESS>and<EMAIL_ADDRESS>

Abstract. For general (1+1)-affine Markov processes, we prove the ergodicity
and exponential ergodicity in total variation distances. Our methods follow
the arguments of ergodic properties for Lévy-driven OU-processes and a
coupling of CBI-processes constructed by stochastic equations driven by time-
space noises. Then the strong Feller property is considered.

Key words: Affine Markov processes; Ergodicity; Strong Feller property; Total
variation distances

##  1 Introduction

Let $m\geq 0$ and $n\geq 0$ be integers. A time-homogeneous Markov processes
$\\{X_{t}:t\geq 0\\}=\\{(Y_{t},Z_{t}):t\geq 0\\}$ taking values in
$\mathbb{G}:=\mathbb{R}_{+}^{m}\times\mathbb{R}^{n}$ is called an affine
Markov process if its characteristic function satisfies

$\displaystyle\mathbb{E}_{x}[\mathrm{e}^{\langle
X_{t},u\rangle}]=\exp\Big{\\{}\langle
x,V(t,u)\rangle+\int_{0}^{t}\psi(V(s,u))\,\mathrm{d}s\Big{\\}},\qquad
x\in\mathbb{G},u\in i\mathbb{R}^{m+n},$ (1.1)

where $V$ and $\psi$ are two complex-valued functions and $V$ satisfies
certain generalized Riccati equations. The affine property means roughly that
the logarithm of the characteristic function is affine with respect to the
initial state. The concept of affine Markov processes enables a unified
treatment of two important Markov classes including continuous state branching
processes with immigration (CBI-processes) and Ornstein-Uhlenbeck type
processes (OU-type processes), where the OU-type processes also include Lévy
processes as a particular case. Roughly speaking, affine processes with state
space $\mathbb{R}_{+}^{m}$ are $m$-dimensional CBI-processes, and those with
state space $\mathbb{R}^{n}$ are $n$-dimensional OU-type processes. The
processes involve rich common mathematical structures and have found
interesting connections and applications in several areas. The general theory
of finite-dimensional affine Markov processes including several equivalent
characterizations and common financial applications was given by Duffie et al.
(2003) under a regularity assumption, which requires the functions

$t\mapsto V(t,u)\quad\mbox{and}\quad
t\mapsto\int_{0}^{t}\psi(V(s,u))\,\mathrm{d}s$

are differentiable at $t=0$ and continuous at $u=0$. The regularity problem
asks whether this property holds automatically for stochastically continuous
affine processes. This property was established in Dawson and Li (2006) under
the first moment condition. The problem was finally settled in Keller-Ressel
et al. (2011), where it was proved that any stochastically continuous affine
Markov process is regular. The connection of the regularity problem with
Hilbert’s fifth problem was also explained in Keller-Ressel et al. (2011).

The ergodicity and strong Feller property of CBI- and OU-type processes have
been studied by a number of authors. In particular, a sufficient and necessary
integrability condition for the ergodicity of a one-dimensional subcritical or
critical CBI-process was announced in Pinsky (1972); see Li (2011) for a
proof. It was proved in Sato and Yamazato (1984) that a finite-dimensional OU-
type process is ergodic if and only if the eigenvalues of its coefficient
matrix have strictly negative real parts. The coupling property and strong
Feller property of finite-dimensional OU-type processes was studied in Priola
and Zabczyk (2009) and Wang (2011). The ergodicity and exponential ergodicity
of such processes in total variation distances were proved in Schilling and
Wang (2012) and Wang (2012). The strong Feller property and exponential
ergodicity in the total variation distance of one-dimensional CBI-processes
were shown in Li and Ma (2015) by a coupling method; see also Li (2020a). In
the recent work of Li (2020b), the ergodicities and exponential ergodicities
in Wasserstein and total variation distances of Dawson-Watanabe superprocesses
with or without immigration were proved, which clearly includes the finite-
dimensional CBI-processes.

For general finite-dimensional affine Markov processes with the strictly
negative real parts of eigenvalues of its coefficient matrix, a sufficient
condition for ergodicity in weak convergence was given in Jin et al. (2020).
The necessity of the condition was not established in Jin et al. (2020), so
their result partially covers the those in Pinsky (1972), Li (2011) and Sato
and Yamazato (1984). The exponential ergodicity of finite-dimensional affine
Markov processes in the Wasserstein distance was established in Friesen et al.
(2020). Zhang and Glynn (2018) provided sufficient conditions for ergodicity
and exponential ergodicity of such processes in the total variation distance.

The main purpose of this paper is to study the ergodicity of the finite-
dimensional affine processes. For simplicity, we focus on the
(1+1)-dimensional affine processes. We prove some results on the ergodicity
and exponential ergodicity of the processes in total variation distances under
natural conditions. Instead of Zhang and Glynn (2018), our approach is based
on coupling methods developed in Schilling and Wang (2012) and Wang (2012);
see also Li and Ma (2015), which answers the question appeared on Jin et al.
(2017), pp.1145; see also Friesen and Jin (2020), pp.646.

The remainder of this paper is organized as follows. In section 2, we prove
the ergodicity of (1+1)-dimensional affine processes in total variance
distances. The exponential ergodic property in total variance distances is
established under stronger conditions in section 3. Finally, the strong Feller
property is studied in section 4.

##  2 Ergodicity in total variance distances

In this and next section, we mainly prove the ergodicity and exponential
ergodicity of (1+1)-affine Markov processes in total variance distances by
dividing the progress of two processes issued from different points into two
parts by a natural coupling of CBI-processes. It is well known that the
coupling property along with the existence of a stationary measure can yield
the ergodicity for the process; see, e.g., Wang (2012), which motivates the
basic proof in this section, as well as for section 3. More precisely, we
prove the coupling property for the process in Proposition 2.3 under proper
conditions, and show the existence of a stationary distribution in Lemma 2.4.
Based on those results, we prove the ergodic property in Theorem 2.5.

We first give some notations. Write
$\mathbb{G}=\mathbb{R}_{+}\times\mathbb{R}$. Define $U=\mathbb{C}_{-}\times
i\mathbb{R}$, where
$\mathbb{C}_{-}=\\{a+ib:a\in\mathbb{R}_{-},b\in\mathbb{R}\\}$ and
$i\mathbb{R}=\\{ia:a\in\mathbb{R}\\}.$ We further define two functions on $U$
as follows:

$\displaystyle\phi(u)=-a_{1}u_{1}-b_{1}u_{2}+(\alpha_{11}+\alpha_{12})u_{1}^{2}+2(\sqrt{\alpha_{11}\alpha_{21}}+\sqrt{\alpha_{12}\alpha_{22}})u_{1}u_{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(\alpha_{21}+\alpha_{22})u_{2}^{2}+\int_{\mathbb{G}}(\mathrm{e}^{\langle
u,z\rangle}-1-\langle u,z\rangle)\,m(\mathrm{d}z),$
$\displaystyle\psi(u)=a_{2}u_{1}-b_{0}u_{2}+\frac{1}{2}\sigma^{2}u_{2}^{2}+\int_{\mathbb{G}}(\mathrm{e}^{\langle
u,z\rangle}-1-z_{2}u_{2})\,n(\mathrm{d}z),$

where $a_{1},b_{i}\in\mathbb{R}$ $(i=0,1,2)$, $a_{2},\sigma\in\mathbb{R}_{+}$,
$(\alpha_{ij})_{2\times 2}$ is a nonnegative matrix, $m$ and $n$ are two Lévy
measures supported on $\mathbb{G}\setminus\\{0\\}$ satisfying

$\displaystyle\int_{\mathbb{G}}\Big{(}z_{1}\wedge
z_{1}^{2}+|z_{2}|^{2}\Big{)}\,m(\mathrm{d}z)+\int_{\mathbb{G}}\Big{(}1\wedge
z_{1}+|z_{2}|\wedge|z_{2}|^{2}\Big{)}\,n(\mathrm{d}z)<\infty.$ (2.1)

Denote by $\\{P_{t}:t\geq 0\\}$ the transition semigroup of (1+1)-affine
Markov process $\\{X_{t}:t\geq 0\\}$. It is well known that $\\{P_{t}:t\geq
0\\}$ can be uniquely determined by

$\int_{\mathbb{G}}\mathrm{e}^{\langle
u,\xi\rangle}\,P_{t}(x,\mathrm{d}\xi)=\exp\Big{\\{}\langle
x,V(t,u)\rangle+\int_{0}^{t}\psi(V(s,u))\,\mathrm{d}s\Big{\\}},$

where

$\left\\{\begin{array}[]{ll}{\frac{\partial V_{1}}{\partial
t}(t,u)=\phi(V(t,u)),}&V_{1}(0,u)=u_{1}.\\\ &\\\
{V_{2}(t,u)=\mathrm{e}^{-b_{2}t}u_{2}}.&\end{array}\right.$

We can also obtain the process as a unique strong solution to a stochastic
integral equation system. Let
$(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P})$ be a filtered probability
space satisfying the usual hypotheses. Denote by
$W_{i}(\mathrm{d}s,\mathrm{d}u),$ $i=0,1,2$ the $(\mathcal{F}_{t})$-Gaussian
white noises on $(0,\infty)^{2}$ with intensity $\mathrm{d}s\mathrm{d}u$,
$M(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)$ be an $(\mathcal{F}_{t})$-Poisson
random measure on $(0,\infty)^{2}\times\mathbb{G}$ with intensity
$\mathrm{d}s\mathrm{d}um(\mathrm{d}z)$ and $N(\mathrm{d}s,\mathrm{d}z)$ an
$(\mathcal{F}_{t})$-Poisson random measure on $(0,\infty)\times\mathbb{G}$
with intensity $\mathrm{d}sn(\mathrm{d}z)$, the corresponding compensated
measures are defined by $\tilde{M}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)$ and
$\tilde{N}(\mathrm{d}s,\mathrm{d}z)$. Let
$P_{i}^{M}(t)=\int_{0}^{t}\int_{0}^{\infty}\int_{\mathbb{G}}\,z_{i}\,M(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)$
and
$P_{i}^{N}(t)=\int_{0}^{t}\int_{\mathbb{G}}\,z_{i}\,N(\mathrm{d}s,\mathrm{d}z)$
for $i=1,2$. We assume those random elements are independent of each other.
Let $Y_{0},Z_{0}$ be $\mathcal{F}_{0}$-measurable random variables and
$Y_{0}\geq 0$. Let us consider the following stochastic integral equation
system:

$\displaystyle
Y_{t}=Y_{0}+\int_{0}^{t}(a_{2}-a_{1}Y_{s})\,\mathrm{d}s+\sqrt{2\alpha_{11}}\int_{0}^{t}\int_{0}^{Y_{s}}\,W_{1}(\mathrm{d}s,\mathrm{d}u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{2\alpha_{12}}\int_{0}^{t}\int_{0}^{Y_{s}}\,W_{2}(\mathrm{d}s,\mathrm{d}u)+\int_{0}^{t}\int_{\mathbb{G}}z_{1}\,N(\mathrm{d}s,\mathrm{d}z)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int_{0}^{t}\int_{0}^{Y_{s-}}\int_{\mathbb{G}}z_{1}\,\tilde{M}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z),$
(2.2) $\displaystyle
Z_{t}=Z_{0}-\int_{0}^{t}(b_{0}+b_{1}Y_{s}+b_{2}Z_{s})\,\mathrm{d}s+\sigma\int_{0}^{t}\int_{0}^{1}\,W_{0}(\mathrm{d}s,\mathrm{d}u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{2\alpha_{21}}\int_{0}^{t}\int_{0}^{Y_{s}}\,W_{1}(\mathrm{d}s,\mathrm{d}u)+\sqrt{2\alpha_{22}}\int_{0}^{t}\int_{0}^{Y_{s}}\,W_{2}(\mathrm{d}s,\mathrm{d}u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int_{0}^{t}\int_{\mathbb{G}}z_{2}\,\tilde{N}(\mathrm{d}s,\mathrm{d}z)+\int_{0}^{t}\int_{0}^{Y_{s-}}\int_{\mathbb{G}}z_{2}\,\tilde{M}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z).$
(2.3)

Here and in the sequel, we understand that for any $a\leq b\in\mathbb{R}$

$\int_{a}^{b}=\int_{(a,b]}\quad\mathrm{and}\quad\int_{a}^{\infty}=\int_{(a,\infty)}.$

The existence of the solution to (2.2)–(2.3) is a consequence of Theorem 6.2
in Dawson and Li (2006), where a weakly equivalent stochastic equation system
was studied. The pathwise uniqueness for (2.2)–(2.3) follows by modifications
of the proofs in Dawson and Li (2006, 2012). Denote by $\\{X_{t}:t\geq
0\\}=\\{(Y_{t},Z_{t}):t\geq 0\\}$ the unique strong solution to (2.2)–(2.3).
Then $\\{Y_{t}:t\geq 0\\}$ is a one-dimensional CBI-process and
$\\{X_{t}:t\geq 0\\}$ is an $(1+1)$-dimensional affine Markov process.

For $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})\in\mathbb{G}$, let
$\\{X_{t}(x):t\geq 0\\}$ and $\\{X_{t}(y):t\geq 0\\}$ be the affine processes
defined by (2.2)–(2.3) starting from $x$ and $y$, respectively. Let
$\varsigma=\inf\\{t\geq 0:Y_{t}(x_{1})=Y_{t}(y_{1})\\}$ be the coalescence
time of the coupling $\\{(Y_{t}(x_{1}),Y_{t}(y_{1})):t\geq 0\\}$. Given $f\in
B_{b}(\mathbb{G}),$ one can see that

$\displaystyle|P_{t}f(x)-P_{t}f(y)|$ $\displaystyle\leq$
$\displaystyle\mathbb{P}\Big{\\{}|f(X_{t}(x))-f(X_{t}(y))|\mathbf{1}_{\\{t<\varsigma\\}}\Big{\\}}$
(2.4)
$\displaystyle+\Big{|}\mathbb{P}\\{[f(X_{t}(x))-f(X_{t}(y))]\mathbf{1}_{\\{t\geq\varsigma\\}}\\}\Big{|}$
$\displaystyle\leq$ $\displaystyle 2\|f\|\mathbb{P}\Big{\\{}Y_{t}(x_{1})\neq
Y_{t}(y_{1})\Big{\\}}$
$\displaystyle+\Big{|}\mathbb{P}\\{[f(X_{t}(x))-f(X_{t}(y))]\mathbf{1}_{\\{t\geq\varsigma\\}}\\}\Big{|}.$

For any $\varepsilon>0,$ we define a finite measure $n_{\varepsilon}$ on
$\mathbb{R}$ such that

$n_{\varepsilon}(B)=\left\\{\begin{array}[]{ll}{n(\mathbb{R}_{+}\times
B)~{},}&\mbox{ if }\,n(\mathbb{R}_{+}\times\mathbb{R})<\infty;\\\ &\\\
{n(\mathbb{R}_{+}\times B_{\varepsilon})~{},}&\mbox{ if
}\,n(\mathbb{R}_{+}\times\mathbb{R})=\infty,\end{array}\right.$

where $B\in\cal{B}(\mathbb{R})$ and
$B_{\varepsilon}:=B\setminus\\{z_{2}:|z_{2}|<\varepsilon\\}$.

In the following we give some key estimates. For $x=(x_{1},x_{2})$ and
$y=(y_{1},y_{2})\in\mathbb{G}$, let $\\{X_{t}(x):t\geq 0\\}$ and
$\\{X_{t}(y):t\geq 0\\}$ be the affine processes defined by (2.2)–(2.3)
starting from $x$ and $y$, respectively. In view of (2.2)–(2.3) we have

$\displaystyle
Z_{t}(x)=T_{t}x_{2}-\int_{0}^{t}T_{t-s}\big{[}b_{0}+b_{1}Y_{s}(x_{1})\big{]}\,\mathrm{d}s+\sqrt{2\alpha_{21}}\int_{0}^{t}\int_{0}^{Y_{s}(x_{1})}T_{t-s}\,W_{1}(\mathrm{d}s,\mathrm{d}u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{2\alpha_{22}}\int_{0}^{t}\int_{0}^{Y_{s}(x_{1})}T_{t-s}\,W_{2}(\mathrm{d}s,\mathrm{d}u)+\sigma\int_{0}^{t}\int_{0}^{1}T_{t-s}\,W_{0}(\mathrm{d}s,\mathrm{d}u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int_{0}^{t}\int_{0}^{Y_{s-}(x_{1})}\int_{\mathbb{G}}T_{t-s}z_{2}\,\tilde{M}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)+\int_{0}^{t}\int_{\mathbb{G}}T_{t-s}z_{2}\,\tilde{N}(\mathrm{d}s,\mathrm{d}z),$
(2.5)

and

$\displaystyle
Z_{t}(x)-Z_{t}(y)=T_{t}(x_{2}-y_{2})-b_{1}\int_{0}^{t}T_{t-s}\Big{(}Y_{s}(x_{1})-Y_{s}(y_{1})\Big{)}\,\mathrm{d}s$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{2\alpha_{21}}\int_{0}^{t}\int_{Y_{s}(y_{1})}^{Y_{s}(x_{1})}T_{t-s}\,W_{1}(\mathrm{d}s,\mathrm{d}u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{2\alpha_{22}}\int_{0}^{t}\int_{Y_{s}(y_{1})}^{Y_{s}(x_{1})}T_{t-s}\,W_{2}(\mathrm{d}s,\mathrm{d}u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\int_{0}^{t}\int_{Y_{s-}(y_{1})}^{Y_{s-}(x_{1})}\int_{\mathbb{G}}T_{t-s}z_{2}\,\tilde{M}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z),$
(2.6)

where $T_{t}=\mathrm{e}^{-b_{2}t}$ for $t\geq 0$.

###### Lemma 2.1

Suppose $0<2b_{2}<a_{1}$. Then there exist strictly positive constants $C_{1}$
and $C_{2}$ such that for any $x,y\in\mathbb{G}$

$\displaystyle\mathbb{E}|Z_{t}(x)-Z_{t}(y)|\leq
T_{t}\Big{(}|x_{2}-y_{2}|+C_{2}(x_{1}-y_{1})+\sqrt{C_{1}(x_{1}-y_{1})}\Big{)},$
(2.7)
$\displaystyle\mathbb{P}\\{|Z_{t}(x)-Z_{t}(y)|>T_{t}\eta\\}\leq\frac{1}{\eta}\Big{(}|x_{2}-y_{2}|+C_{2}(x_{1}-y_{1})+\sqrt{C_{1}(x_{1}-y_{1})}\Big{)}$
(2.8)

for $\eta>0$.

Proof. Note that
$\mathbb{E}[Y_{s}(x_{1})-Y_{s}(y_{1})]=(x_{1}-y_{1})\mathrm{e}^{-a_{1}s}.$ By
Martingale inequality and Cauchy-Schwartz inequality we see that

$\displaystyle\mathbb{E}\Big{|}\int_{0}^{t}\int_{Y_{s-}(y_{1})}^{Y_{s-}(x_{1})}\int_{\mathbb{G}}\mathrm{e}^{b_{2}s}z_{2}\,\tilde{M}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)\Big{|}\leq\bigg{[}\mathbb{E}\Big{|}\int_{0}^{t}\int_{Y_{s-}(y_{1})}^{Y_{s-}(x_{1})}\int_{\mathbb{G}}\mathrm{e}^{b_{2}s}z_{2}\,\tilde{M}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z)\Big{|}^{2}\bigg{]}^{\frac{1}{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq\bigg{[}\frac{\mathrm{e}^{(2b_{2}-a_{1})t}-1}{2b_{2}-a_{1}}(x_{1}-y_{1})\int_{\mathbb{G}}|z_{2}|^{2}\,m(\mathrm{d}z)\bigg{]}^{\frac{1}{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq\sqrt{C_{11}(x_{1}-y_{1})},$

similarly,

$\displaystyle\mathbb{E}\Big{|}\sqrt{2\alpha_{21}}\int_{0}^{t}\int_{Y_{s}(y_{1})}^{Y_{s}(x_{1})}\mathrm{e}^{b_{2}s}\,W_{1}(\mathrm{d}s,\mathrm{d}u)+\sqrt{2\alpha_{22}}\int_{0}^{t}\int_{Y_{s}(y_{1})}^{Y_{s}(x_{1})}\mathrm{e}^{b_{2}s}\,W_{2}(\mathrm{d}s,\mathrm{d}u)\Big{|}$
$\displaystyle\leq\bigg{[}2\alpha_{21}\frac{\mathrm{e}^{(2b_{2}-a_{1})t}-1}{2b_{2}-a_{1}}(x_{1}-y_{1})\bigg{]}^{\frac{1}{2}}+\bigg{[}2\alpha_{22}\frac{\mathrm{e}^{(2b_{2}-a_{1})t}-1}{2b_{2}-a_{1}}(x_{1}-y_{1})\bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq\sqrt{C_{12}(x_{1}-y_{1})},$

and

$\mathbb{E}\Big{|}-b_{1}\int_{0}^{t}\mathrm{e}^{b_{2}s}(Y_{s-}(x_{1})-Y_{s-}(y_{1}))\,\mathrm{d}s\Big{|}=\Big{|}\frac{b_{1}(\mathrm{e}^{(b_{2}-a_{1})t}-1)}{b_{2}-a_{1}}(x_{1}-y_{1})\Big{|}\leq
C_{2}(x_{1}-y_{1}),$

where
$C_{11}=\frac{1}{a_{1}-2b_{2}}\int_{\mathbb{G}}|z_{2}|^{2}\,m(\mathrm{d}z),C_{12}=8\max\Big{\\{}\frac{\alpha_{21}}{a_{1}-2b_{2}},\frac{\alpha_{22}}{a_{1}-2b_{2}}\Big{\\}}$
and $C_{2}=\frac{|b_{1}|}{a_{1}-b_{2}}.$ By (2) it is not hard to see that

$\displaystyle\mathbb{E}|Z_{t}(x)-Z_{t}(y)|\leq
T_{t}\Big{(}|x_{2}-y_{2}|+C_{2}(x_{1}-y_{1})+\sqrt{C_{1}(x_{1}-y_{1})}\Big{)},$

where $C_{1}=4\max\\{C_{11},C_{12}\\}.$ The second inequality is an immediate
result following from Markov inequality and (2.7). $\Box$

For a bounded measurable function $f$ on $\mathbb{R}$, define the supremum
norm $\|f\|=\sup_{x}|f(x)|$. Given two bounded measures $\mu$ and $\nu$ on
$(\mathbb{R},\cal{B}(\mathbb{R}))$, let
$\mu\wedge\nu=\mu-(\mu-\nu)^{+}=\nu-(\nu-\mu)^{+}$, where the superscript
”$+$” refers to the positive part in the Jordan-Hahn decomposition. It is easy
to see that $\mu\wedge\nu=\nu\wedge\mu=2^{-1}(\mu+\nu-|\mu-\nu|)$, where
$|\mu-\nu|=(\mu-\nu)^{+}+(\nu-\mu)^{+}$ is the total variation measure. Let
$\|\cdot\|_{\mathrm{var}}$ denote the total variation norm defined by

$\|\mu-\nu\|_{\mathrm{var}}=\sup_{\|f\|\leq 1}|\mu(f)-\nu(f)|,$

where $\mu(f)=\int f\mathrm{d}\mu$. For the convenience, we formulate the
following conditions:

$(\textbf{A})$ Denote the branching mechanism of the CBI-processes
$\\{Y_{t}:t\geq 0\\}$ by

$\phi_{0}(x)=a_{1}x+(\alpha_{11}+\alpha_{12})x^{2}+\int_{\mathbb{G}}(\mathrm{e}^{-xz_{1}}-1+xz_{1})\,m(\mathrm{d}z),\quad
x\geq 0.$

There exists $\theta>0$ such that for any $z\geq\theta,\phi_{0}(z)>0$, and

$\int_{\theta}^{\infty}\phi_{0}^{-1}(z)\,\mathrm{d}z<\infty.$

$(\textbf{B})$ There exist two constants $\varepsilon,~{}\eta>0$ such that

$\displaystyle\inf_{|a|\leq\eta}n_{\varepsilon}\wedge(\delta_{a}\ast
n_{\varepsilon})(\mathbb{R})>0,\quad\int_{\\{|z_{2}|>\varepsilon\\}}|z_{2}|\,n(\mathrm{d}z)<\infty.$

###### Remark 2.2

(1). Condition (A) is called Grey’s condition; see, e.g., Grey (1974). This
condition has been used to study the exponentially ergodic property in total
variance distances and strong Feller property of one-dimensional subcritical
CBI-processes; see,e.g., Li and Ma (2015) and Li (2020a).

(2). Condition (B) is sharp to study the coupling property for the Lévy
process $(L_{t})_{t\geq 0}$ with Lévy measure n and the Ornstein-Uhlenbeck
process driven by $(L_{t})_{t\geq 0}$. Intuitively, it is one possibility to
guarantee the sufficient jump activity such that the process admits a
successful coupling; see,e.g., Schilling and Wang (2011,2012) and Wang (2012).

###### Proposition 2.3

Suppose that Conditions (A,B) hold and $0<2b_{2}<a_{1}$. Then there exist
constants $\hat{C},\kappa>0$, such that for any $t>0,x,y\in\mathbb{G},$ we
have

$\|P_{t}(x,\cdot)-P_{t}(y,\cdot)\|_{\mathrm{var}}\leq\hat{C}\Big{(}1+(\bar{v}_{\frac{1}{\kappa^{2}+1}t}+1)|x_{1}-y_{1}|+\sqrt{|x_{1}-y_{1}|}+|x_{2}-y_{2}|\Big{)}\frac{1}{\sqrt{t}},$

where $\bar{v}_{t}$ is the unique solution of the following equation:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bar{v}_{t}=-\phi_{0}(\bar{v}_{t}),\quad
t>0$ (2.9)

with initial condition $\bar{v}_{0+}=\infty$. Moreover, the mapping
$t\mapsto\bar{v}_{t}$ is decreasing.

Proof. Under Condition (A), the solution $\bar{v}_{t}$ to (2.9) is unique, and
the mapping $t\mapsto\bar{v}_{t}$ is decreasing; see, e.g., Theorem 3.6,
Theorem 3.7 and Corollary 3.14 in Li (2020a). Under Condition (B), for
simplicity, we let

$L_{t}^{\varepsilon}=\int_{0}^{t}\int_{\\{|z_{2}|>\varepsilon\\}}z_{2}\,N(\mathrm{d}s,\mathrm{d}z)$

and define a sequence of stopping times
$\Upsilon^{\varepsilon}_{n}=\inf\\{t>\Upsilon^{\varepsilon}_{n-1}:L_{t}^{\varepsilon}\neq
L_{t-}^{\varepsilon}\\}$ with convention $\Upsilon^{\varepsilon}_{0}=0$. For
$i\geq 1$ let
$\tau^{\varepsilon}_{i}=\Upsilon^{\varepsilon}_{i}-\Upsilon^{\varepsilon}_{i-1}$
and
$U^{\varepsilon}_{i}=\int_{\\{\Upsilon^{\varepsilon}_{i}\\}}\int_{\\{|z_{2}|>\varepsilon\\}}z_{2}\,N(\mathrm{d}s,\mathrm{d}z).$
Then $(\tau^{\varepsilon}_{i})_{i\geq 1}$ are i.i.d. random variables which
are exponentially distributed with intensity
$C_{\varepsilon}=n_{\varepsilon}(\mathbb{R})$ and
$(U^{\varepsilon}_{i})_{i\geq 1}$ of i.i.d. random variables on $\mathbb{R}$
with distribution $\bar{n}_{\varepsilon}:=n_{\varepsilon}/C_{\varepsilon}$.
Moreover, the two sequences $(U^{\varepsilon}_{i})_{i\geq 1}$ and
$(\tau^{\varepsilon}_{i})_{i\geq 1}$ are independent of each other. Let
$N^{\varepsilon}_{t}=\sup\\{k\geq 1:\sum_{i=1}^{k}\tau^{\varepsilon}_{i}\leq
t\\}$. Then $(N^{\varepsilon}_{t})_{t\geq 0}$ is a Poisson process of
intensity $C_{\varepsilon}$. Now we can rewrite

$\int_{0}^{t}\int_{\\{|z_{2}|>\varepsilon\\}}z_{2}\,N(\mathrm{d}s,\mathrm{d}z)=\sum\limits_{i=1}^{N^{\varepsilon}_{t}}U^{\varepsilon}_{i}$

with $\sum_{i=1}^{0}=0$ by convention. It is not hard to check that

$\displaystyle\int_{0}^{t}\int_{\\{|z_{2}|\geq\varepsilon\\}}T_{t-s}z_{2}\,N(\mathrm{d}s,\mathrm{d}z)=0\cdot\mathbf{1}_{\\{\Upsilon^{\varepsilon}_{1}>t\\}}+\sum_{k=1}^{\infty}\mathbf{1}_{\\{\Upsilon^{\varepsilon}_{k}\leq
t<\Upsilon^{\varepsilon}_{k+1}\\}}\sum_{i=1}^{k}T_{t-\Upsilon^{\varepsilon}_{i}}U^{\varepsilon}_{i}.$

Let us use the following notations for the convenience,

$\displaystyle\zeta_{t,\varepsilon}=T_{t}\Big{\\{}(b_{0}-\int_{\\{|z_{2}|>\varepsilon\\}}z_{2}\,n(\mathrm{d}z))\int_{0}^{t}\mathrm{e}^{b_{2}s}\,\mathrm{d}s$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sigma\int_{0}^{t}\int_{0}^{1}\mathrm{e}^{b_{2}s}\,W_{0}(\mathrm{d}s,\mathrm{d}u)+\int_{0}^{t}\int_{\\{|z_{2}|\leq\varepsilon\\}}\mathrm{e}^{b_{2}s}z_{2}\,\tilde{N}(\mathrm{d}s,\mathrm{d}z)\Big{\\}},$
$\displaystyle\theta_{t}(x_{1})=-b_{1}\int_{0}^{t}\mathrm{e}^{b_{2}s}Y_{s}(x_{1})\,\mathrm{d}s+\int_{0}^{t}\int_{0}^{Y_{s-}(x_{1})}\int_{\mathbb{G}}\mathrm{e}^{b_{2}s}z_{2}\,\tilde{M}(\mathrm{d}s,\mathrm{d}u,\mathrm{d}z),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{2\alpha_{21}}\int_{0}^{t}\int_{0}^{Y_{s}(x_{1})}\mathrm{e}^{b_{2}s}\,W_{1}(\mathrm{d}s,\mathrm{d}u)+\sqrt{2\alpha_{22}}\int_{0}^{t}\int_{0}^{Y_{s}(x_{1})}\mathrm{e}^{b_{2}s}\,W_{2}(\mathrm{d}s,\mathrm{d}u).$

Moreover, for any $t\geq 0$, we denote the distribution of
$\theta_{t}(x_{1})-\theta_{t}(y_{1})$ by $\Gamma_{t,x_{1}-y_{1}}$. In view of
(2.3), for any $f\in B_{b}(\mathbb{G})$,

$\displaystyle\mathbb{E}f(Y_{t}(x_{1}),Z_{t}(x))=\mathbb{E}f\Big{(}Y_{t}(x_{1}),T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta_{t,\varepsilon}+\sum^{N^{\varepsilon}_{t}}_{k=1}T_{t-\Upsilon^{\varepsilon}_{k}}U^{\varepsilon}_{k}\Big{)}$
$\displaystyle=\mathbb{E}\Big{[}f\Big{(}Y_{t}(x_{1}),T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta_{t,\varepsilon}\Big{)}\mathbf{1}_{\\{N^{\varepsilon}_{t}=0\\}}\Big{]}$
$\displaystyle~{}~{}+\mathbb{E}f\Big{(}Y_{t}(x_{1}),\sum_{k=1}^{\infty}\mathbf{1}_{\\{\Upsilon^{\varepsilon}_{k}\leq
t<\Upsilon^{\varepsilon}_{k+1}\\}}\Big{\\{}T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta_{t,\varepsilon}+\sum_{j=1}^{k}T_{\Upsilon^{\varepsilon}_{j}}U^{\varepsilon}_{j}\Big{\\}}\Big{)}$
$\displaystyle=\mathbb{E}\Big{[}f\Big{(}Y_{t}(x_{1}),T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta_{t,\varepsilon}\Big{)}\mathbf{1}_{\\{N^{\varepsilon}_{t}=0\\}}\Big{]}$
$\displaystyle~{}~{}+\sum_{k=1}^{\infty}\int...\int_{\sum_{i=1}^{k}t_{i}\leq
t<\sum_{i=1}^{k+1}t_{i}}C^{k+1}_{\varepsilon}\mathrm{e}^{-C_{\varepsilon}\sum_{i=1}^{k+1}t_{i}}\,\mathrm{d}t_{1}\,...\mathrm{d}t_{k+1}$
$\displaystyle~{}~{}\times\int_{\mathbb{R}^{k}}\mathbb{E}f\Big{(}Y_{t}(x_{1}),T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta_{t,\varepsilon}+\sum_{i=1}^{k}T_{\sum_{j=1}^{i}t_{j}}r_{i}\Big{)}\,\bar{n}_{\varepsilon}(\mathrm{d}r_{1})\,...\bar{n}_{\varepsilon}(\mathrm{d}r_{k})$
$\displaystyle=\mathbb{E}\Big{[}f\Big{(}Y_{t}(x_{1}),T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta_{t,\varepsilon}\Big{)}\mathbf{1}_{\\{N^{\varepsilon}_{t}=0\\}}\Big{]}$
$\displaystyle~{}~{}+\sum_{k=1}^{\infty}\int...\int_{\sum_{i=1}^{k}t_{i}\leq
t<\sum_{i=1}^{k+1}t_{i}}C^{k+1}_{\varepsilon}\mathrm{e}^{-C_{\varepsilon}\sum_{i=1}^{k+1}t_{i}}\,\mathrm{d}t_{1}\,...\mathrm{d}t_{k+1}$
$\displaystyle~{}~{}\quad\times\int_{\mathbb{R}}\mathbb{E}f\Big{(}Y_{t}(x_{1}),T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta_{t,\varepsilon}+z\Big{)}\,n_{t_{1},...,t_{k}}(\mathrm{d}z),$

where the second equality partly follows from formula (2.10) in Schilling and
Wang (2012); see also Lemma 2.2 in Schilling and Wang (2012). Here,
$n_{t_{1},...t_{k}}(\mathrm{d}z)$ is the probability measure on $\mathbb{R}$,
which is the image of the $k$-fold product measure
$\bar{n}_{\varepsilon}\times...\times\bar{n}_{\varepsilon}$ under the linear
transformation $J_{t_{1},...,t_{k}}:$
$J_{t_{1},...t_{k}}(r_{1},...,r_{k})=T_{t_{1}}r_{1}+...+T_{t_{1}+...+t_{k}}r_{k}.$
Given $(x_{1},x_{2}),(y_{1},y_{2})\in\mathbb{G},$ without loss of generality,
we can assume that $x_{1}\geq y_{1}$, and we have

$\displaystyle\sup_{\|f\|\leq
1}\Big{|}\mathbb{E}\big{[}(f(X_{t}(x))-f(X_{t}(y)))\mathbf{1}_{\\{t\geq\varsigma\\}}\big{]}\Big{|}$
$\displaystyle\leq
2\mathrm{e}^{-C_{\varepsilon}t}+\sum_{k=1}^{\infty}\int...\int_{\sum_{i=1}^{k}t_{i}\leq
t<\sum_{i=1}^{k+1}t_{i}}C^{k+1}_{\varepsilon}\mathrm{e}^{-C_{\varepsilon}\sum_{i=1}^{k+1}t_{i}}\,\mathrm{d}t_{1}\,...\mathrm{d}t_{k+1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\sup_{\|f\|\leq
1}\Big{|}\int_{\mathbb{R}}\mathbb{E}f\Big{(}Y_{t}(x_{1}),T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta_{t,\varepsilon}+z\Big{)}\,n_{t_{1},...,t_{k}}(\mathrm{d}z)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\int_{\mathbb{R}}\mathbb{E}f\Big{(}Y_{t}(y_{1}),T_{t}(y_{2}+\theta_{t}(y_{1}))+\zeta_{t,\varepsilon}+z\Big{)}\,n_{t_{1},...,t_{k}}(\mathrm{d}z)\Big{|}$
$\displaystyle\leq
2\mathrm{e}^{-C_{\varepsilon}t}+\sum_{k=1}^{\infty}\int...\int_{\sum_{i=1}^{k}t_{i}\leq
t<\sum_{i=1}^{k+1}t_{i}}C^{k+1}_{\varepsilon}\mathrm{e}^{-C_{\varepsilon}\sum_{i=1}^{k+1}t_{i}}\,\mathrm{d}t_{1}\,...\mathrm{d}t_{k+1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\sup_{\|f\|\leq
1}\int_{\mathbb{R}}\,\Big{|}\int_{\mathbb{R}}\mathbb{E}f(Y_{t}(y_{1}),T_{t}(y_{2}+\theta_{t}(y_{1}))+\zeta_{t,\varepsilon}+z)\,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\qquad\qquad\times\delta_{T_{t}(x_{2}-y_{2}+z_{2})}\ast
n_{t_{1},...,t_{k}}(\mathrm{d}z)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\int_{\mathbb{R}}\mathbb{E}f(Y_{t}(y_{1}),T_{t}(y_{2}+\theta_{t}(y_{1}))+\zeta_{t,\varepsilon}+z\Big{)}\,n_{t_{1},...,t_{k}}(\mathrm{d}z)\Big{|}\,\Gamma_{t,x_{1}-y_{1}}(\mathrm{d}z_{2})$
$\displaystyle\leq
2\mathrm{e}^{-C_{\varepsilon}t}+\sum_{k=1}^{\infty}\int...\int_{\sum_{i=1}^{k}t_{i}\leq
t<\sum_{i=1}^{k+1}t_{i}}C^{k+1}_{\varepsilon}\mathrm{e}^{-C_{\varepsilon}\sum_{i=1}^{k+1}t_{i}}\,\mathrm{d}t_{1}\,...\mathrm{d}t_{k+1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\int_{W}\varpi(z_{2},t,t_{1},...,t_{k})\,\Gamma_{t,x_{1}-y_{1}}(\mathrm{d}z_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2\sum_{k=1}^{\infty}\int...\int_{\sum_{i=1}^{k}t_{i}\leq
t<\sum_{i=1}^{k+1}t_{i}}C^{k+1}_{\varepsilon}\mathrm{e}^{-C_{\varepsilon}\sum_{i=1}^{k+1}t_{i}}\,\mathrm{d}t_{1}\,...\mathrm{d}t_{k+1}\int_{\mathbb{R}\setminus
W}\,\Gamma_{t,x_{1}-y_{1}}(\mathrm{d}z_{2}),$

where $W=\\{z_{2}:|z_{2}+x_{2}-y_{2}|<\eta\\}$ for some $\eta>0$ and

$\displaystyle\varpi(z_{2},t,t_{1},...,t_{k})=\|\delta_{T_{t}(x_{2}-y_{2}+z_{2})}\ast
n_{t_{1},...,t_{k}}-n_{t_{1},...,t_{k}}\|_{\mathrm{var}}.$

If we set $p(t,x,y):=\int_{\mathbb{R}\setminus
W}\,\Gamma_{t,x_{1}-y_{1}}(\mathrm{d}z_{2}),$ one can see that

$\displaystyle\sum_{k=1}^{\infty}\int...\int_{\sum_{i=1}^{k}t_{i}\leq
t<\sum_{i=1}^{k+1}t_{i}}C^{k+1}_{\varepsilon}\mathrm{e}^{-C_{\varepsilon}\sum_{i=1}^{k+1}t_{i}}\,\mathrm{d}t_{1}\,...\mathrm{d}t_{k+1}\int_{\mathbb{R}\setminus
W}\,\Gamma_{t,x_{1}-y_{1}}(\mathrm{d}z_{2})$ $\displaystyle\leq
p(t,x,y)\mathrm{e}^{-C_{\varepsilon}t}\sum_{k=1}^{\infty}C^{k+1}_{\varepsilon}\int...\int_{t\geq\sum_{i=1}^{k}t_{i}}\,\mathrm{d}t_{1}\,...\mathrm{d}t_{k}$
$\displaystyle=p(t,x,y)C_{\varepsilon}(1-\mathrm{e}^{-C_{\varepsilon}t}).$

By (2), we
have$T_{t}^{-1}[Z_{t}(x)-Z_{t}(y)]=(x_{2}-y_{2})+\theta_{t}(x_{1})-\theta_{t}(y_{1}).$
It follows that
$T_{t}^{-1}|Z_{t}(x)-Z_{t}(y)|=|x_{2}-y_{2}+\theta_{t}(x_{1})-\theta_{t}(y_{1}))|$.
Then

$\displaystyle p(t,x,y)\leq\mathbb{P}\\{|Z_{t}(x)-Z_{t}(y)|>T_{t}\eta\\}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq\frac{1}{\eta}\Big{(}|x_{2}-y_{2}|+C_{2}(x_{1}-y_{1})+\sqrt{C_{1}(x_{1}-y_{1})}\Big{)},$
(2.10)

where the last inequality follows from (2.8). On the other hand, following the
proof of Theorem 1.1 in Schilling and Wang (2012), for
$|x_{2}-y_{2}+z_{2}|\leq\eta,t\geq t_{1}+...+t_{k}$ and $k\geq 1$ we can find
some constants $C_{1,\eta},C_{3}>0$ such that

$\varpi(z_{2},t,t_{1},...,t_{k})\leq\frac{C_{1,\eta}}{\sqrt{k}},$

and

$\displaystyle\frac{C_{3}}{\sqrt{t}}\geq
2\mathrm{e}^{-C_{\varepsilon}t}+\sum_{k=1}^{\infty}\int...\int_{\sum_{i=1}^{k}t_{i}\leq
t<\sum_{i=1}^{k+1}t_{i}}C^{k+1}_{\varepsilon}\mathrm{e}^{-C_{\varepsilon}\sum_{i=1}^{k+1}t_{i}}\,\mathrm{d}t_{1}\,...\mathrm{d}t_{k+1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\int_{W}\varpi(z_{2},t,t_{1},...,t_{k})\,\Gamma_{t,x_{1}-y_{1}}(\mathrm{d}z_{2}).$

In conclusion, we have

$\displaystyle\Big{|}\mathbb{E}\big{[}(f(X_{t}(x))-f(X_{t}(y)))\mathbf{1}_{\\{t\geq\varsigma\\}}\big{]}\Big{|}\leq\frac{\|f\|C_{3}}{\sqrt{t}}+\frac{2\|f\|C_{\varepsilon}}{\eta}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\qquad\qquad\qquad\qquad\quad\times\Big{(}|x_{2}-y_{2}|+C_{2}(x_{1}-y_{1})+\sqrt{C_{1}(x_{1}-y_{1})}\Big{)}.$

In view of (2.4), for $f\in B_{b}(\mathbb{G})$

$\displaystyle|P_{t}f(x)-P_{t}f(y)|\leq
2\|f\|\bar{v}_{t}(x_{1}-y_{1})+\frac{\|f\|C_{3}}{\sqrt{t}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{2\|f\|C_{\varepsilon}}{\eta}\Big{(}|x_{2}-y_{2}|+C_{2}(x_{1}-y_{1})+\sqrt{C_{1}(x_{1}-y_{1})}\Big{)}.$

For any $s\in(0,t),$ by using the Markov property and (2.7),

$\displaystyle|P_{t}f(x)-P_{t}f(y)|$
$\displaystyle=\Big{|}\mathbb{E}\Big{\\{}P_{s}f(Y_{t-s}(x_{1}),Z_{t-s}(x))-P_{s}f(Y_{t-s}(y_{1}),Z_{t-s}(y))\Big{\\}}\Big{|}$
$\displaystyle\leq
2\|f\|\bar{v}_{s}\mathbb{E}(Y_{t-s}(x_{1})-Y_{t-s}(y_{1}))+\frac{\|f\|C_{3}}{\sqrt{s}}$
$\displaystyle~{}~{}~{}~{}+\frac{2\|f\|C_{\varepsilon}}{\eta}\Big{\\{}\mathbb{E}|Z_{t-s}(x)-Z_{t-s}(y)|+\Big{[}C_{1}\mathbb{E}\\{Y_{t-s}(x_{1})-Y_{t-s}(y_{1})\\}\Big{]}^{\frac{1}{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+C_{2}\mathbb{E}\\{Y_{t-s}(x_{1})-Y_{t-s}(y_{1})\\}\Big{\\}}$
$\displaystyle\leq
2\|f\|\bar{v}_{s}(x_{1}-y_{1})\mathrm{e}^{-a_{1}(t-s)}+\frac{\|f\|C_{3}}{\sqrt{s}}$
$\displaystyle~{}~{}~{}~{}+\frac{2\|f\|C_{\varepsilon}}{\eta}\Big{\\{}\sqrt{C_{1}(x_{1}-y_{1})\mathrm{e}^{-a_{1}(t-s)}}+C_{2}(x_{1}-y_{1})\mathrm{e}^{-a_{1}(t-s)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mathrm{e}^{-b_{2}(t-s)}(|x_{2}-y_{2}|+C_{2}(x_{1}-y_{1})+\sqrt{C_{1}(x_{1}-y_{1})})\Big{\\}},$

which implies that

$\displaystyle\|P_{t}(x,\cdot)-P_{t}(y,\cdot)\|_{\mathrm{var}}$
$\displaystyle\leq$ $\displaystyle
C_{4}\Big{[}1+(\bar{v}_{s}+1)(x_{1}-y_{1})+\sqrt{x_{1}-y_{1}}+|x_{2}-y_{2}|\Big{]}\Big{(}\mathrm{e}^{-b_{2}(t-s)}\vee\frac{1}{\sqrt{s}}\Big{)}$
$\displaystyle\leq$ $\displaystyle
C_{4}\Big{[}1+(\bar{v}_{s}+1)(x_{1}-y_{1})+\sqrt{x_{1}-y_{1}}+|x_{2}-y_{2}|\Big{]}\Big{(}\frac{\kappa}{\sqrt{t-s}}\vee\frac{1}{\sqrt{s}}\Big{)}$

for some constants $C_{4}>0$ and $\kappa=1\vee\frac{1}{\mathrm{e}b_{2}}.$ We
finally obtain the required assertion by setting
$s=\frac{1}{\kappa^{2}+1}t,\,\,\hat{C}=\sqrt{\kappa^{2}+1}C_{4}.$ $\Box$

Note that the initial assumptions of Lévy measures $m$ and $n$ in Jin et al.
(2020) are weaker than (2.1). Before we establish the main result in this
section, we need the following lemma, which is a consequence of Theorem 2.7 in
Jin et al. (2020).

###### Lemma 2.4

Assume that $a_{1}>0,b_{2}>0$, (2.1) and

$\displaystyle\int_{\\{z_{1}\geq 1\\}}\log z_{1}\,n(\mathrm{d}z)<\infty$
(2.11)

hold. Then the law of $X_{t}$ converges weakly to a limiting distribution
$\pi$ given by

$\int_{\mathbb{G}}\mathrm{e}^{\langle
u,y\rangle}\,\pi(\mathrm{d}y)=\exp\Big{\\{}\int_{0}^{\infty}\psi(V(s,u))\,\mathrm{d}s\Big{\\}},\quad
u\in U,$

Moreover, $\pi$ is the unique stationary distribution for $X.$

Based on the Lemma 2.4 and Proposition 2.3 above, we can prove the ergodicity
of the transition semigroup $(P_{t})_{t\geq 0}.$ More explicitly, we have the
following result:

###### Theorem 2.5

Suppose $\int_{\\{z_{1}\geq 1\\}}\log z_{1}\,n(\mathrm{d}z)<\infty$ and the
conditions of Proposition 2.3 are satisfied. Then the affine process
$(X_{t})_{t\geq 0}$ is ergodic in the total variation distance. Namely, there
exists a unique invariant measure $\pi$ for the process such that for any
$x\in\mathbb{G},$

$\displaystyle\lim\limits_{t\to\infty}\|P_{t}(x,\cdot)-\pi(\cdot)\|_{\mathrm{var}}=0.$

Proof. By (2.1) and Lemma 2.4, there exists a unique invariant measure $\pi$
for $(P_{t})_{t\geq 0}.$ Fix $x\in\mathbb{G},$ one can see that

$\|P_{t}(x,\cdot)-\pi(\cdot)\|_{\mathrm{var}}\leq\int_{\mathbb{G}}\|P_{t}(x,\cdot)-P_{t}(y,\cdot)\|_{\mathrm{var}}\,\pi(\mathrm{d}y).$

For any $\epsilon>0,$ we choose $\delta>0$ such that
$\pi\\{|x_{1}-y_{1}|+|x_{2}-y_{2}|>\delta\\}\leq\epsilon.$ By Proposition 2.3
we have

$\displaystyle\|P_{t}(x,\cdot)-\pi(\cdot)\|_{\mathrm{var}}$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{G}\setminus\Xi}\|P_{t}(x,\cdot)-P_{t}(y,\cdot)\|_{\mathrm{var}}\,\pi(\mathrm{d}y)+\int_{\Xi}\|P_{t}(x,\cdot)-P_{t}(y,\cdot)\|_{\mathrm{var}}\,\pi(\mathrm{d}y)$
$\displaystyle\leq$
$\displaystyle\frac{\hat{C}}{\sqrt{t}}\int_{\mathbb{G}\setminus\Xi}\Big{(}1+(\bar{v}_{\frac{1}{\kappa^{2}+1}t}+1)\delta+\sqrt{\delta}+\delta\Big{)}\,\pi(\mathrm{d}y)+2\pi(\Xi)$
$\displaystyle\leq$
$\displaystyle\frac{\hat{C}}{\sqrt{t}}\int_{\mathbb{G}\setminus\Xi}\Big{(}1+(\bar{v}_{\frac{1}{\kappa^{2}+1}t}+1)\delta+\sqrt{\delta}+\delta\Big{)}\,\pi(\mathrm{d}y)+2\epsilon,$

where $\Xi=\\{y:|x_{1}-y_{1}|+|x_{2}-y_{2}|>\delta\\}.$ Letting $t\to\infty$
first and then $\epsilon\to 0,$ we prove the desired result. $\Box$

##  3 Exponential ergodicity in total variance distances

In this section, we prove the exponential ergodicity of (1+1)-affine Markov
processes defined by (2.2)–(2.3) with (2.1) in total variance distance under
proper conditions. Since the exponential ergodicity implies the ergodicity,
it’s reasonable to strengthen the assumptions to guarantee the exponential
ergodicity. We also give the following conditions before moving forward:

$(\textbf{C})$ There exists $\varepsilon>0,$ such that

$\displaystyle\limsup\limits_{\rho\to
0}\Big{[}\frac{\sup_{|a|\leq\rho}\|n_{\varepsilon}-\delta_{a}\ast
n_{\varepsilon}\|_{\mathrm{var}}}{\rho}\Big{]}<\infty,\quad\int_{\\{|z_{2}|>\varepsilon\\}}|z_{2}|\,n(\mathrm{d}z)<\infty.$

$(\textbf{C}^{\prime})$ There exists $\varepsilon>0,$ such that

$\displaystyle\limsup\limits_{\rho\to
0}\Big{[}\frac{\sup_{|a|\leq\rho}\|n_{\varepsilon}-\delta_{a}\ast
n_{\varepsilon}\|_{\mathrm{var}}}{\rho}\Big{]}<\infty,\quad\int_{\\{|z_{2}|>1\\}}|z_{2}|^{2}\,n(\mathrm{d}z)<\infty.$

###### Remark 3.1

Condition (C’) plays an important role in characterizing the exponential
ergodicity. This condition implies
$\int_{\mathbb{G}}|z_{2}|^{2}\,n(\mathrm{d}z)<\infty$, so it is stronger than
(C). Condition (B) is weaker than (C); see, e.g., Remark 1 in Wang (2012) for
a proof.

###### Proposition 3.2

Suppose that Conditions (A,C) hold and $0<2b_{2}<a_{1}$. Then there exist
constants $\tilde{C},\tilde{\kappa}>0,$ such that for any
$t>0,x,y\in\mathbb{G},$ we have

$\|P_{t}(x,\cdot)-P_{t}(y,\cdot)\|_{\mathrm{var}}\leq\tilde{C}\Big{(}1+(\bar{v}_{\frac{\tilde{\kappa}t}{C_{\varepsilon}}}+1)|x_{1}-y_{1}|+\sqrt{|x_{1}-y_{1}|}+|x_{2}-y_{2}|\Big{)}\mathrm{e}^{-\tilde{\kappa}t},$

where $\bar{v}_{t}$ is defined as that in Proposition 2.3.

Proof. Following the arguments in the proof of Proposition 2.3, it implies
that on $\\{N^{\varepsilon}_{t}\geq 1\\}$

$\int_{0}^{t}\int_{\\{|z_{2}|\geq\varepsilon\\}}T_{t-s}z_{2}\,N(\mathrm{d}s,\mathrm{d}z)=\sum_{k=1}^{N^{\varepsilon}_{t}}T_{t-\Upsilon^{\varepsilon}_{k}}U^{\varepsilon}_{k}.$

For given $f\in B_{b}(\mathbb{G}),x\in\mathbb{G}$ we have the following
decomposition

$\mathbb{E}[f\left(Y_{t}(x_{1}),Z_{t}(x)\right)]=\mathbb{E}[f(Y_{t}(x_{1}),Z_{t}(x))\mathbf{1}_{\\{N^{\varepsilon}_{t}=0\\}}]+P^{1}_{t}f(x),$

where
$P^{1}_{t}f(x)=\mathbb{E}[f(Y_{t}(x_{1}),Z_{t}(x))\mathbf{1}_{\\{N^{\varepsilon}_{t}\geq
1\\}}]$ and

$\displaystyle
P^{1}_{t}f(x)=\mathbb{E}\Big{\\{}\mathbf{1}_{\\{N^{\varepsilon}_{t}\geq
1\\}}f\Big{(}Y_{t}(x_{1}),T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta^{1}_{t,\varepsilon}+T_{t-\Upsilon^{\varepsilon}_{N^{\varepsilon}_{t}}}U^{\varepsilon}_{N^{\varepsilon}_{t}}\Big{)}\Big{\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{1}{C_{\varepsilon}}\mathbb{E}\Big{\\{}\mathbf{1}_{\\{N^{\varepsilon}_{t}\geq
1\\}}\int_{\mathbb{R}}f\Big{(}Y_{t}(x_{1}),T_{t}(x_{2}+\theta_{t}(x_{1}))+\zeta^{1}_{t,\varepsilon}+T_{t-\Upsilon^{\varepsilon}_{N^{\varepsilon}_{t}}}z\Big{)}\,n_{\varepsilon}(\mathrm{d}z)\Big{\\}},$

where

$\displaystyle\zeta^{1}_{t,\varepsilon}=T_{t}\Big{\\{}(b_{0}-\int_{\\{|z_{2}|>\varepsilon\\}}z_{2}\,n(\mathrm{d}z))\int_{0}^{t}\mathrm{e}^{b_{2}s}\,\mathrm{d}s+\sigma\int_{0}^{t}\int_{0}^{1}\mathrm{e}^{b_{2}s}\,W_{0}(\mathrm{d}s,\mathrm{d}u)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{k=1}^{N^{\varepsilon}_{t}-1}T_{-\Upsilon^{\varepsilon}_{k}}U^{\varepsilon}_{k}+\int_{0}^{t}\int_{\\{|z_{2}|\leq\varepsilon\\}}\mathrm{e}^{b_{2}s}z_{2}\,\tilde{N}(\mathrm{d}s,\mathrm{d}z)\Big{\\}}.$

We recall again that the distributions of
$\theta_{t}(x_{1})-\theta_{t}(y_{1})$ is $\Gamma_{t,x_{1}-y_{1}}$. Given
$x,y\in\mathbb{G}$, without loss of generality, it suffices to consider the
case of $x_{1}\geq y_{1}.$

$\displaystyle\Big{|}\mathbb{E}[(f(X_{t}(x)-f(X_{t}(y))))\mathbf{1}_{\\{N^{\varepsilon}_{t}\geq
1\\}}\mathbf{1}_{\\{t\geq\varsigma\\}}]\Big{|}$
$\displaystyle\leq\frac{1}{C_{\varepsilon}}\Big{|}\mathbb{E}\Big{\\{}\mathbf{1}_{\\{N^{\varepsilon}_{t}\geq
1\\}}\Big{(}\int_{\mathbb{R}}f(z_{1},t,y_{1},y_{2})\,n_{\varepsilon}\Big{(}\mathrm{d}z_{1}-T_{\Upsilon^{\varepsilon}_{N^{\varepsilon}_{t}}}(x_{2}-y_{2}+z_{2})\Big{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}\times\,\int_{\mathbb{R}}\,\Gamma_{t,x_{1}-y_{1}}(\mathrm{d}z_{2})-\int_{\mathbb{R}}f(z_{1},t,y_{1},y_{2})\,n_{\varepsilon}(\mathrm{d}z_{1})\Big{)}\Big{\\}}\Big{|}$
$\displaystyle\leq\frac{(1-\mathrm{e}^{-C_{\varepsilon}t})\Lambda\,\|f\|}{C_{\varepsilon}}\int_{\mathbb{R}}\Big{(}|x_{2}-y_{2}|+|z_{2}|\Big{)}\,\Gamma_{t,x_{1}-y_{1}}(\mathrm{d}z_{2}),$

where

$\displaystyle\Lambda=\sup\limits_{\rho>0}\Big{[}\frac{\sup_{|a|\leq\rho}\|n_{\varepsilon}-\delta_{a}\ast
n_{\varepsilon}\|_{\mathrm{var}}}{\rho}\Big{]},$ $\displaystyle
f(z_{1},t,y_{1},y_{2})=f\Big{(}Y_{t}(y_{1}),T_{t}(y_{2}+\theta_{t}(y_{1}))+\zeta^{1}_{t,\varepsilon}+T_{t-\Upsilon^{\varepsilon}_{N^{\varepsilon}_{t}}}z_{1}\Big{)}.$

We have $\Lambda<\infty$ due to the Condition (C) and the fact that
$\sup\limits_{|a|\leq\rho}\|n_{\varepsilon}-\delta_{a}\ast
n_{\varepsilon}\|_{\mathrm{var}}\leq 2C_{\varepsilon}.$ It together with Lemma
2.1 follow that

$\Big{|}\mathbb{E}[(f(X_{t}(x))-f(X_{t}(y)))\mathbf{1}_{\\{N^{\varepsilon}_{t}\geq
1\\}}\mathbf{1}_{\\{t\geq\varsigma\\}}]\Big{|}\leq\frac{\Lambda\|f\|}{C_{\varepsilon}}\Big{\\{}|x_{2}-y_{2}|+\sqrt{C_{1}(x_{1}-y_{1})}+C_{2}(x_{1}-y_{1})\Big{\\}}.$

On the other hand,

$\mathbb{E}\Big{|}f(Y_{t}(x_{1}),Z_{t}(x))\mathbf{1}_{\\{N^{\varepsilon}_{t}=0\\}}\mathbf{1}_{\\{t\geq\varsigma\\}}\Big{|}\leq\|f\|\mathrm{e}^{-C_{\varepsilon}t},\quad
t\geq 0,f\in B_{b}(\mathbb{G}),$

it follows that

$\displaystyle\Big{|}\mathbb{E}[(f(X_{t}(x))-f(X_{t}(y)))\mathbf{1}_{\\{t\geq\varsigma\\}}]\Big{|}$
$\displaystyle\leq
2\|f\|\mathrm{e}^{-C_{\varepsilon}t}+\frac{\Lambda}{C_{\varepsilon}}\,\|f\|\Big{\\{}|x_{2}-y_{2}|+\sqrt{C_{1}(x_{1}-y_{1})}+C_{2}(x_{1}-y_{1})\Big{\\}}.$

Therefore, it together with (2.4) and Theorem 10.3 in Li (2020a) imply that

$\displaystyle|P_{t}f(x)-P_{t}f(y)|\leq
2\|f\|(\bar{v}_{t}(x_{1}-y_{1})+\mathrm{e}^{-C_{\varepsilon}t})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\|f\|\Big{\\{}\frac{\Lambda}{C_{\varepsilon}}(|x_{2}-y_{2}|+\sqrt{C_{1}(x_{1}-y_{1})}+C_{2}(x_{1}-y_{1}))\Big{\\}}.$

Similarly as that in Proposition 2.3, for any $s\in(0,t),$ by using the Markov
property and (2.7) we have

$\displaystyle|P_{t}f(x)-P_{t}f(y)|$
$\displaystyle=\Big{|}\mathbb{E}\Big{[}P_{s}f(Y_{t-s}(x_{1}),Z_{t-s}(x))-P_{s}f(Y_{t-s}(y_{1}),Z_{t-s}(y))\Big{]}\Big{|}$
$\displaystyle\leq
2\|f\|\bar{v}_{s}\mathbb{E}[Y_{t-s}(x_{1})-Y_{t-s}(y_{1})]+2\|f\|\mathrm{e}^{-C_{\varepsilon}s}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{\|f\|\Lambda}{C_{\varepsilon}}\Big{\\{}\mathbb{E}|Z_{t-s}(x)-Z_{t-s}(y)|+\Big{\\{}C_{1}\mathbb{E}[Y_{t-s}(x_{1})-Y_{t-s}(y_{1})]\Big{\\}}^{\frac{1}{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+C_{2}\mathbb{E}[Y_{t-s}(x_{1})-Y_{t-s}(y_{1})]\Big{\\}}$
$\displaystyle\leq
2\|f\|\bar{v}_{s}(x_{1}-y_{1})\mathrm{e}^{-a_{1}(t-s)}+2\|f\|\mathrm{e}^{-C_{\varepsilon}s}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\|f\|\Lambda}{C_{\varepsilon}}\Big{\\{}\mathrm{e}^{-b_{2}(t-s)}\Big{(}|x_{2}-y_{2}|+C_{2}(x_{1}-y_{1})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{C_{1}(x_{1}-y_{1})}\Big{)}+[C_{1}(x_{1}-y_{1})\mathrm{e}^{-a_{1}(t-s)}]^{\frac{1}{2}}+C_{2}(x_{1}-y_{1})\mathrm{e}^{-a_{1}(t-s)}\Big{\\}},$

it follows that

$\|P_{t}(x,\cdot)-P_{t}(y,\cdot)\|_{\mathrm{var}}\leq\tilde{C}\Big{(}1+(\bar{v}_{s}+1)(x_{1}-y_{1})+\sqrt{x_{1}-y_{1}}+|x_{2}-y_{2}|\Big{)}\Big{\\{}\mathrm{e}^{-C_{\varepsilon}s}\vee\mathrm{e}^{-b_{2}(t-s)}\Big{\\}},$

where
$\tilde{C}=\max\\{2,\frac{\Lambda}{C_{\varepsilon}},\frac{C_{2}\Lambda}{C_{\varepsilon}},\frac{\sqrt{C_{1}}\Lambda}{C_{\varepsilon}}\\}.$
Setting $s=\frac{b_{2}t}{C_{\varepsilon}+b_{2}},$ we obtain

$\|P_{t}(x,\cdot)-P_{t}(y,\cdot)\|_{\mathrm{var}}\leq\tilde{C}\Big{(}1+(\bar{v}_{\frac{\tilde{\kappa}t}{C_{\varepsilon}}}+1)(x_{1}-y_{1})+\sqrt{x_{1}-y_{1}}+|x_{2}-y_{2}|\Big{)}\mathrm{e}^{-\tilde{\kappa}t},$

where $\tilde{\kappa}=\frac{b_{2}C_{\varepsilon}}{C_{\varepsilon}+b_{2}}.$
Then the required assertion holds. $\Box$

Based on Proposition 3.2, we have the following result:

###### Theorem 3.3

Suppose that Conditions (A,C’) hold and $0<2b_{2}<a_{1}$. Moreover, assume
that $\int_{\\{z_{1}>1\\}}z_{1}\,n(\mathrm{d}z)<\infty$ holds. Then the affine
process $(X_{t})_{t\geq 0}$ is exponentially ergodic in the total variation
distance.

Proof. Given $x\in\mathbb{G},$ it follows from Proposition 3.2 that

$\displaystyle\|P_{t}(x,\cdot)-\pi(\cdot)\|_{\mathrm{var}}\leq\int_{M}\|P_{t}(x,\cdot)-P_{t}(y,\cdot)\|_{\mathrm{var}}\,\pi(\mathrm{d}y)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\qquad\quad\quad\leq\tilde{C}\mathrm{e}^{-\tilde{\kappa}t}\Big{(}1+(\bar{v}_{\frac{\tilde{\kappa}t}{C_{\varepsilon}}}+1)(x_{1}+\Delta_{1})+\sqrt{x_{1}+\Delta_{1}}+|x_{2}|+\Delta_{2}\Big{)},$

where

$\Delta_{1}=\int_{\mathbb{G}}y_{1}\,\pi(\mathrm{d}y),\quad\Delta_{2}=\int_{\mathbb{G}}|y_{2}|\,\pi(\mathrm{d}y).$

In order to prove the exponential ergodic property, it suffices to prove
$\Delta_{1}$ and $\Delta_{2}$ are finite. For $u\in U,$ setting
$u=(u_{1},0),u_{1}\leq 0,$ then

$\int_{\mathbb{G}}e^{u_{1}y_{1}}\,\pi(\mathrm{d}y)=\exp\Big{\\{}\int_{0}^{\infty}P(V_{1}(s,u))\,\mathrm{d}s\Big{\\}},\quad\frac{\partial
V_{1}(s,u)}{\partial s}=\tilde{\phi}_{0}(V_{1}(s,u)),$

where $P$ and $\tilde{\phi}_{0}$ are two functions defined on $\mathbb{R}_{-}$
such that

$P(x)=a_{2}x+\int_{\mathbb{G}}(\mathrm{e}^{xz_{1}}-1)\,n(\mathrm{d}z),\quad\tilde{\phi}_{0}(x)=-a_{1}x+(\alpha_{11}+\alpha_{12})x^{2}+\int_{\mathbb{G}}(\mathrm{e}^{xz_{1}}-1-xz_{1})\,m(\mathrm{d}z).$

Following the proof of Theorem 10.4 in Li (2020a), in the case of $a_{1}>0,$
we can see that for $u\in U$, $\lim\limits_{s\to\infty}V_{1}(s,u)=0$ and

$\int_{0}^{t}P(V_{1}(s,u))\,\mathrm{d}s=-\int_{V_{1}(t,u)}^{u_{1}}\frac{P(z)}{\tilde{\phi}_{0}(z)}\,\mathrm{d}z.$

It implies that

$\int_{\mathbb{G}}e^{u_{1}y_{1}}\,\pi(\mathrm{d}y)=\exp\Big{\\{}-\int_{0}^{u_{1}}\frac{P(z)}{\tilde{\phi}_{0}(z)}\,\mathrm{d}z\Big{\\}}.$

By differentiating both sides of above at $\lambda_{1}=0$ we get
$\Delta_{1}=a_{1}^{-1}[a_{2}+\int_{\mathbb{G}}z_{1}\,n(\mathrm{d}z)]<\infty.$
On the other hand, $\\{Z_{t}\\}_{t\geq 0}$ can be constructed by

$Z_{t}=\mathrm{e}^{-b_{2}t}Z_{0}+\int_{0}^{t}\mathrm{e}^{-b_{2}(t-s)}\mathrm{d}L^{Y}_{s},\quad
t\geq 0,$

where

$\displaystyle\mathrm{d}L^{Y}_{t}=-(b_{0}+b_{1}Y_{t})\mathrm{d}t+\sigma\int_{0}^{1}\,W_{0}(\mathrm{d}t,\mathrm{d}u)+\sqrt{2\alpha_{21}}\int_{0}^{Y_{t}}\,W_{1}(\mathrm{d}t,\mathrm{d}u)$
$\displaystyle~{}~{}~{}~{}~{}~{}+\sqrt{2\alpha_{22}}\int_{0}^{Y_{t}}\,W_{2}(\mathrm{d}t,\mathrm{d}u)+\int_{\mathbb{G}}z_{2}\,\tilde{N}(\mathrm{d}t,\mathrm{d}z)+\int_{0}^{Y_{t-}}\int_{\mathbb{G}}z_{2}\,\tilde{M}(\mathrm{d}t,\mathrm{d}u,\mathrm{d}z).$

We assume that $Z_{0}=y_{2},Y_{0}=y_{1}.$ By Cauchy-Schwartz inequality and
the result for the first moment of CBI-processes; see, e.g., Li (2020a),
pp.33, one can see that

$\displaystyle\mathbb{E}|\int_{0}^{t}\mathrm{e}^{b_{2}s}\,\mathrm{d}L^{Y}_{s}|\leq|b_{0}|\frac{\mathrm{e}^{b_{2}t}-1}{b_{2}}+|b_{1}|\int_{0}^{t}\mathrm{e}^{b_{2}s}\Big{(}y_{1}\mathrm{e}^{-a_{1}s}+\frac{\gamma}{a_{1}}(1-\mathrm{e}^{-a_{1}s})\Big{)}\,\mathrm{d}s$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\Big{[}\int_{\mathbb{G}}z_{2}^{2}\,m(\mathrm{d}z)\int_{0}^{t}\mathrm{e}^{2b_{2}s}\Big{(}y_{1}\mathrm{e}^{-a_{1}s}+\frac{\gamma}{a_{1}}(1-\mathrm{e}^{-a_{1}s})\Big{)}\,\mathrm{d}s\Big{]}^{\frac{1}{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\Big{[}\int_{\mathbb{G}}z_{2}^{2}\,n(\mathrm{d}z)(\mathrm{e}^{b_{2}t}-1)b^{-1}_{2}\Big{]}^{\frac{1}{2}},$

where $\gamma=a_{2}+\int_{\mathbb{G}}z_{1}\,n(\mathrm{d}z)$. Then

$\sup_{t>0}\mathbb{E}|\int_{0}^{t}\mathrm{e}^{-b_{2}(t-s)}\,\mathrm{d}L^{Y}_{s}|\leq\frac{2|b_{1}|}{b_{2}-a_{1}}y_{1}+C_{5}+\Big{[}\frac{2\int_{\mathbb{G}}z_{2}^{2}\,m(\mathrm{d}z)}{2b_{2}-a_{1}}y_{1}+C_{6}\Big{]}^{\frac{1}{2}}$

for some constants $C_{5}$ and $C_{6}$. Further, for any $k\geq 1$ and $t>0$,

$\mathbb{E}(|Z_{t}(y)|\wedge k)\leq\mathrm{e}^{-b_{2}t}|y_{2}|\wedge
k+\sup_{t>0}\mathbb{E}|\int_{0}^{t}\mathrm{e}^{-b_{2}(t-s)}\,\mathrm{d}L^{Y}_{s}|,$

and so

$\int_{\mathbb{G}}(|y_{2}|\wedge
k)\,\pi(\mathrm{d}y)\leq\int_{\mathbb{G}}(\mathrm{e}^{-b_{2}t}|y_{2}|\wedge
k)\,\pi(\mathrm{d}y)+C_{7},\quad t>0,\,\,k\geq 1$

for some constant $C_{7}$ since $\Delta_{1}<\infty$. Letting first
$t\to\infty$ and then $k\to\infty$, it follows from dominated convergence
theorem and $b_{2}>0$ that

$\int_{\mathbb{G}}|y_{2}|\,\pi(\mathrm{d}y)<\infty.$

That completes the proof. $\Box$

##  4 Strong Feller property

In this section, we study the strong Feller property of (1+1)-dimensional
affine Markov processes constructed by (2.2)–(2.3). We first formulate the
following condition:

$(\textbf{D})$ There exists a nonnegative measurable function $\varrho_{0}$ on
$\mathbb{R}$ such that

$n(\mathbb{R}_{+}\times\mathrm{d}z_{2})\geq\sigma_{0}(\mathrm{d}z_{2}):=\varrho_{0}(z_{2})\mathrm{d}z_{2},\quad\sigma_{0}(\mathbb{R})>0.$

For a positive integrally function $\mathrm{g}$ defined on $\mathbb{R}$, for
$k\geq 1,$ let
$\sigma_{k}:=(k\mathrm{g}\wedge\rho_{0})(z_{2})\,\mathrm{d}z_{2},$
$\sigma_{k}(\mathbb{R}):=\int_{\mathbb{R}}(k\mathrm{g}\wedge\rho_{0})(z_{2})\,\mathrm{d}z_{2}$.
There exists $K\geq 1$ such that for all $k\geq K$, the measure $\sigma_{k}$
satisfies

$\displaystyle\limsup\limits_{\rho\to
0}\Big{[}\frac{\sup_{|a|\leq\rho}\|\sigma_{k}-\delta_{a}\ast\sigma_{k}\|_{\mathrm{var}}}{\rho}\Big{]}<\infty,\quad\int_{\mathbb{R}}|z_{2}|\,\sigma_{k}(\mathrm{d}z_{2})<\infty.$

Under Condition (D), $\sigma_{k}(\mathbb{R})<\infty$ for $k\geq 1$. Then for
$k\geq K$ we can define a compound Poisson process as follows:

$L_{t}^{k}:=\int_{0}^{t}\int_{\mathbb{G}}z_{2}\,N_{k}(\mathrm{d}s,\mathrm{d}z),$

where $N_{k}(\mathrm{d}s,\mathrm{d}z)$ is a Poisson random measure with
intensity $\mathrm{d}sn_{k}(\mathrm{d}z)$ satisfying
$n_{k}(\mathbb{R}_{+}\times\mathrm{d}z_{2})=\sigma_{k}(\mathrm{d}z_{2})$. We
define a sequence of stopping times
$\Upsilon^{k}_{n}=\inf\\{t>\Upsilon^{k}_{n-1}:L^{k}_{t}\neq L_{t-}^{k}\\}$
with convention $\Upsilon^{k}_{0}=0$. For $i\geq 1$ let
$\tau^{k}_{i}=\Upsilon^{k}_{i}-\Upsilon^{k}_{i-1}$ and
$U^{k}_{i}=\int_{\\{\Upsilon^{k}_{i}\\}}\int_{\mathbb{G}}z_{2}\,N_{k}(\mathrm{d}s,\mathrm{d}z).$
Then it is easy to see that $(\tau^{k}_{i})_{i\geq 1}$ are i.i.d. random
variables which are exponentially distributed with intensity
$\sigma_{k}(\mathbb{R})$ and $(U^{k}_{i})_{i\geq 1}$ are i.i.d. random
variables on $\mathbb{R}$ with distribution
$\sigma_{k}/\sigma_{k}(\mathbb{R})$. Moreover, the two sequences are
independent of each other. Let $N_{t}^{k}=\sup\\{j\geq
1:\sum_{i=1}^{j}\tau_{i}^{k}\leq t\\}$. Then $(N_{t}^{k})_{t\geq 0}$ is a
Poisson process of intensity $\sigma_{k}(\mathbb{R})$. Now we can rewrite

$\int_{0}^{t}\int_{\mathbb{G}}z_{2}\,N_{k}(\mathrm{d}s,\mathrm{d}z)=\sum\limits_{i=1}^{N^{k}_{t}}U^{k}_{i}$

with $\sum_{i=1}^{0}=0$ by convention.

###### Lemma 4.1

Suppose Condition (D) holds. For $k\geq K$ let $P^{k}_{t}$ be defined by

$P^{k}_{t}f(x):=\mathbb{E}\Big{[}f(Y_{t}(x_{1}),Z_{t}(x))\mathbf{1}_{\\{\tau^{k}_{1}\leq
t\\}}\Big{]},\quad f\in B_{b}(\mathbb{G}),\quad x\in\mathbb{G}.$

Then there exist a constant $C_{k,8}>0$ and non-negative functions $t\mapsto
C_{1}(t)$ and $t\mapsto C_{2}(t)$ such that for any given $x,y\in\mathbb{G}$,

$\displaystyle\sup_{\|f\|\leq 1}|P^{k}_{t}f(x)-P^{k}_{t}f(y)|$
$\displaystyle\leq
2\mathrm{e}^{-\sigma_{k}(\mathbb{R})t}+C_{k,8}\Big{\\{}|x_{2}-y_{2}|+\sqrt{C_{1}(t)|x_{1}-y_{1}|}+C_{2}(t)|x_{1}-y_{1}|\Big{\\}}.$

Proof. By a modified proof in Proposition 3.2, we have

$\displaystyle\sup_{\|f\|\leq 1}|P^{k}_{t}f(x)-P^{k}_{t}f(y)|$
$\displaystyle\leq
2\mathrm{e}^{-\sigma_{k}(\mathbb{R})t}+\frac{\Lambda_{k}}{\sigma_{k}(\mathbb{R})}\Big{\\{}|x_{2}-y_{2}|+\sqrt{C_{1}(t)|x_{1}-y_{1}|}+C_{2}(t)|x_{1}-y_{1}|\Big{\\}},$

where

$\displaystyle
C_{1}(t)=\bar{c}\cdot\frac{\mathrm{e}^{(2b_{2}-a_{1})t}-1}{2b_{2}-a_{1}},\quad
C_{2}(t)=\frac{|b_{1}|(\mathrm{e}^{(b_{2}-a_{1})t}-1)}{b_{2}-a_{1}},$
$\displaystyle\bar{c}=4\max\Big{\\{}8(\alpha_{21}\vee\alpha_{22}),\int_{\mathbb{G}}|z_{2}|^{2}\,m(\mathrm{d}z)\Big{\\}},\quad\Lambda_{k}=\sup\limits_{\rho>0}\Big{[}\frac{\sup_{|a|\leq\rho}\|\sigma_{k}-\delta_{a}\ast\sigma_{k}\|_{\mathrm{var}}}{\rho}\Big{]}$

by conventions $C_{1}(t)=\bar{c}t$ for $2b_{2}=a_{1}$ and $C_{2}(t)=|b_{1}|t$
for $b_{2}=a_{1}.$ And we prove the required assertion by setting
$C_{k,8}=\frac{\Lambda_{k}}{\sigma_{k}(\mathbb{R})}.$ $\Box$

###### Theorem 4.2

Suppose Conditions (A,D) hold with $\sigma_{0}(\mathbb{R})=\infty$. Then the
affine Markov process $(X_{t})_{t\geq 0}$ satisfies the strong Feller
property.

Proof. It follows from (2.4) that for $x,y\in\mathbb{G}$ we have

$\displaystyle\sup_{\|f\|\leq 1}|P_{t}f(x)-P_{t}f(y)|\leq
2\bar{v}_{t}|x_{1}-y_{1}|+\sup_{\|f\|\leq
1}|P^{k}_{t}f(x)-P^{k}_{t}f(y)|,\quad k\geq K,$

which together with Lemma 4.1 imply that

$\displaystyle\lim\limits_{y\to x}\sup_{\|f\|\leq 1}|P_{t}f(x)-P_{t}f(y)|\leq
2\mathrm{e}^{-\sigma_{k}(\mathbb{R})t},\quad k\geq K,\quad t>0.$

Since $\sigma_{k}(\mathbb{R})\uparrow\infty$ as $k\uparrow\infty,$ we conclude
the strong Feller property of $(X_{t})_{t\geq 0}$. $\Box$

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