# Mind the Cap! - Constrained Portfolio Optimisation in Heston’s Stochastic
Volatility Model

M. ESCOBAR-ANEL${\dagger}$ M. KSCHONNEK∗${\ddagger}$ and R. ZAGST${\ddagger}$
∗Corresponding author. Email<EMAIL_ADDRESS>${\dagger}$ Department
of Statistical and Actuarial Sciences, Western University, London, ON, Canada
${\ddagger}$ Chair of Mathematical Finance, Technical University of Munich,
D-80333 Munich, Germany

(31st August 2024)

###### Abstract

We consider a portfolio optimisation problem for a utility-maximising investor
who faces convex constraints on his portfolio allocation in Heston’s
stochastic volatility model. We apply the duality methods developed in
Escobar-Anel et al. (2023) to obtain a closed-form expression for the optimal
portfolio allocation. In doing so, we observe that allocation constraints
impact the optimal constrained portfolio allocation in a fundamentally
different way in Heston’s stochastic volatility model than in the Black
Scholes model. In particular, the optimal constrained portfolio may be
different from the naive ‘capped’ portfolio, which caps off the optimal
unconstrained portfolio at the boundaries of the constraints. Despite this
difference, we illustrate by way of a numerical analysis that in most
realistic scenarios the capped portfolio leads to slim annual wealth
equivalent losses compared to the optimal constrained portfolio. During a
financial crisis, however, a capped solution might lead to compelling annual
wealth equivalent losses.

###### keywords:

Portfolio Optimisation, Allocation Constraints, Dynamic Programming, Heston’s
Stochastic Volatility Model, Incomplete Markets

G11, C61

##  1 Introduction

Despite its widespread use in both academia and the financial industry, it has
been well-documented in the mathematical finance literature that a variety of
properties of financial time series, so-called stylised facts, are not
captured by the Black-Scholes model (see e.g. Cont (2001), Lux (2009),
Weatherall (2018)). A major point of criticism is the constant volatility of
modelled log returns, whereas empirical returns appear to be time-dependent
and random (Taylor (1994)). The stochastic volatility model proposed by Heston
(1993) aims to avoid this gap by modelling the volatility of log returns as a
Cox-Ingersoll-Ross process (CIR process). While the Heston model was
originally proposed in the context of option pricing, its analytical
tractability has led to insightful applications in continuous-time portfolio
optimisation (see e.g. Kraft (2005)), which is the subject of this article.
Specifically, we consider the portfolio optimisation problem of an investor
trading continuously in a financial market consisting of one risk-free asset
and one risky asset with stochastic Heston volatility. The investor seeks to
maximise his expected utility derived from terminal wealth at a finite time
point $T>0$ under the condition that his portfolio allocation $\pi$ abides by
given convex allocation constraints $K.$ The considered optimisation problem
involves two major facets, which differ from the original portfolio
optimisation setting of Merton (1971): stochasticity of the volatility of
risky asset log returns (‘stochastic volatility’) and the presence of convex
allocation constraints. The following paragraphs present a brief overview of
the relevant literature, with respect to both of these facets.

_(i) Portfolio Optimisation in Financial Markets with Stochastic Volatility._

Our continuous time set-up can be traced back to the seminal work of Merton
(1971), who used dynamic programming methods to derive explicit solutions to
the unconstrained dynamic portfolio optimisation problem for an investor with
HARA utility function in a Black-Scholes model with constant volatility.
Generalisations for different utility functions and more complex financial
markets were achieved by employing martingale methods (Pliska (1986) and
Karatzas et al. (1987)), which rely heavily on the assumption that all
contingent claims in the financial market are replicable. However, this
assumption is not generally satisfied for financial markets with stochastic
volatility, which means that martingale methods are not directly applicable,
unless the financial market is artificially completed by the addition of
fictitious, volatility-dependent assets (see e.g. Liu and Pan (2003), Branger
et al. (2008), Egloff et al. (2010), Escobar-Anel et al. (2017), and Chen et
al. (2021)). Without such completion, the solvability of the portfolio
optimisation problem in financial markets with stochastic volatility is often
directly linked to the solvability of the associated HJB PDE.111Note that
obtaining and formally verifying the optimality of a candidate portfolio
process requires more than just a solution to the associated HJB PDE, as
pointed out by Korn and Kraft (2004). Solutions to such portfolio optimisation
problems were first characterised in terms of viscosity solutions to the
associated HJB PDE for multi-factor models in Zariphopoulou (2001), and
explicit closed-form solutions for Heston’s stochastic volatility model were
first derived and formally verified in Kraft (2005). Subsequently, Liu (2006)
derived explicit solution formulae for an optimal consumption and portfolio
allocation problem in a general multi-factor model, where the factor is a
quadratic diffusion. Kallsen and Muhle-Karbe (2010) used the notion of an
opportunity process and semi-martingale characteristics to develop an approach
which leads to closed-form solutions for the optimal portfolio process in a
range of exponentially affine stochastic volatility models, including the
Heston model and the jump model of Geman et al. (2003). The PCSV model and the
Wishart process were considered as multi-dimensional extensions of the Heston
model for multidimensional asset universes with stochastic correlation in
Escobar-Anel et al. (2017) and Bäuerle and Li (2013). An extensive overview of
related papers in the field of dynamic portfolio optimisation in stochastic
factor models can be found in Zariphopoulou (2009).

_(ii) Portfolio Optimisation in Financial Markets with Convex Allocation
Constraints._

Convex allocation constraints were first considered in the context of
continuous-time dynamic portfolio optimisation in Karatzas et al. (1991) and
Cvitanic and Karatzas (1992). The authors showed that a solution to the primal
constrained portfolio optimisation problem could be derived by considering a
family of ‘auxiliary’ unconstrained portfolio optimisation problems,
determining the optimal portfolios for these problems, and selecting the least
favourable portfolio among them. For CRRA-utility functions, Cvitanic and
Karatzas (1992) determined the optimal constrained portfolio process in
closed-form up to a deterministic minimiser of a real convex optimisation
problem. Zariphopoulou (1994) considered a similar setting in a one-
dimensional Black Scholes market, but did not employ any duality techniques.
Rather, the author characterised the value function as the unique viscosity
solution of the associated HJB PDE and gave a semi-explicit expression of the
optimal portfolio allocation in terms of the value function. From both
Cvitanic and Karatzas (1992) and Zariphopoulou (1994), one can easily see that
the optimal constrained portfolio allocation for an investor with a CRRA
utility function in the Black-Scholes model is equal to the unconstrained
optimal solution if the constraints are satisfied, and is otherwise capped at
the boundary of the constraint. Ever since, allocation constraints have been
integrated into portfolio optimisation problems in a myriad of ways (see e.g.
Cuoco (1997), Pham (2002), Lindberg (2006), Mnif (2007), Bian et al. (2011),
Nutz (2012) and Dong and Zheng (2020)). Regardless, explicit solutions for the
optimal portfolio allocation were obtained only on rare occasions. The
availability of explicit expressions appears to strongly depend on the
interplay between the chosen financial model, utility function and constraint.
In this spirit, Escobar-Anel et al. (2023) recently developed a duality
approach, which enabled them to state a condition under which the
exponentially affine separability structure of the value function (as
discussed in Liu (2006)) is retained under the addition of convex allocation
constraints. If this condition is satisfied, then a candidate for the optimal
constrained portfolio allocation is known up to the solution of a Riccati ODE
and a deterministic minimiser of a convex optimisation problem. Further, the
authors showed that this condition is satisfied for Heston’s stochastic
volatility model, but failed to determine explicit solutions to the associated
ODEs and did not formally verify the optimality of the candidate portfolio
allocation.

In this paper, we make a threefold contribution to this literature:

  * •

We complement the work of Escobar-Anel et al. (2023) by deriving the first
explicit, closed-form expression for the optimal portfolio allocation
$\pi^{\ast}$ in Heston’s stochastic volatility model under the presence of
convex allocation constraints. This optimality is verified formally in a
verification theorem.

  * •

We show that the optimal portfolio allocation $\pi^{\ast}$ may be different
from the capped optimal unconstrained portfolio allocation $\pi_{u}$ for
Heston’s stochastic volatility model. In particular, we prove an equivalent
characterisation which describes when these two portfolio are different. We
conduct a numerical study to show that this difference is slim for calm market
scenarios, but can lead to significant annual wealth equivalent losses during
turbulent market scenarios.

  * •

We extend these results from the one-dimensional Heston’s stochastic
volatility model to a multi-dimensional PCSV model with constraints on the
exposures to individual stochastic market factors and to generalised financial
markets with inverse volatility constraints.

The remainder of this paper is structured as follows: In Section 2, we
introduce and solve the constrained portfolio optimisation problem
$\mathbf{(P)}$ in Heston’s stochastic volatility model. Specifically, we
derive a solution to the HJB PDE associated with $\mathbf{(P)}$ and the
candidate optimal portfolio $\pi^{\ast}$ in Section 2.1, verify its optimality
formally by proving a verification theorem in Section 2.2 and discuss its
relation to the optimal unconstrained portfolio $\pi_{u}$ in Section 2.3. In
Section 3, we consider a generalised financial market model which depends on a
multi-dimensional CIR process and derive the optimal portfolio in the PCSV
model under exposure constraints (Section 3.1) and in general financial
markets with inverse volatility constraints (Section 3.2). In Section 4, we
illustrate our theoretical results for Heston’s stochastic volatility model in
a numerical analysis, where we analyse the wealth equivalent loss of the
optimal constrained portfolio for a Black Scholes model (Section 4.1) and the
capped optimal unconstrained portfolio for Heston’s stochastic volatility
model (Section 4.2). Section 5 concludes the paper. All proofs relating to
this paper can be found in Appendix A.

##  2 Heston’s Stochastic Volatility Model

We consider a finite time horizon $T>0$ and a complete, filtered probability
space $(\Omega,\mathcal{F}_{T},\mathbbm{F}=(\mathcal{F}_{t})_{t\in[0,T]},Q)$,
in which the filtration $\mathbbm{F}$ is generated by two independent Wiener
processes
$\big{(}\hat{W},W^{z})=\big{(}\hat{W}(t),W^{z}(t)\big{)}_{t\in[0,T]}.$ We
define a financial market $\mathcal{M}_{H}$, consisting of one risk-free asset
$P_{0}$ and one risky asset $P_{1}$, where the risky asset’s instantaneous
variance is driven by a CIR process $z$. Specifically, we assume a market
consisting of one risk-free asset $P_{0}$ and one risky asset $P_{1}$, which
satisfy $P_{0}(0)=P_{1}(0)=1$ and follow the dynamics

$\displaystyle dP_{0}(t)$ $\displaystyle=P_{0}(t)rdt,$ $\displaystyle
dP_{1}(t)$ $\displaystyle=P_{1}(t)\big{(}r+\eta\cdot
z(t)\big{)}dt+\sqrt{z(t)}\underbrace{\left(\rho
dW^{z}(t)+\sqrt{1-\rho^{2}}d\hat{W}(t)\right)}_{=:dW(t)},$

where $z(0)=z_{0}>0$ and

$\displaystyle dz(t)$
$\displaystyle=\kappa(\theta-z(t))dt+\sigma\sqrt{z(t)}dW^{z}(t).$

The market coefficients $r,\eta,\kappa,\theta,\sigma,z_{0}$ are assumed to be
positive constants, and $\rho\in(-1,1).$ Lastly, it is assumed that Feller’s
condition holds for the parameters of $z$, i.e. $2\kappa\theta>\sigma^{2},$
and, therefore, $z(t)$ is guaranteed to take only positive values with
probability 1 (see Gikhman (2011)). The wealth process $V^{v_{0},\pi}$ of an
investor trading in $\mathcal{M}_{H}$ according to a relative portfolio
process $\pi$ and initial wealth $v_{0}>0$ now satisfies the usual SDE

$\displaystyle dV^{v_{0},\pi}(t)=V^{v_{0},\pi}(t)\Big{(}[r+(\eta
z(t)\pi(t)]dt+\pi(t)\cdot\sqrt{z(t)}dW(t)\Big{)}.$ (1)

In this context, the portfolio process $\pi=\big{(}\pi(t)\big{)}_{t\in[0,T]}$
is one-dimensional, and represents the fraction of wealth invested in the
risky asset $P_{1}$ at time $t$. The remaining fraction $1-\pi(t)$ is invested
in the risk-free asset $P_{0}$. We restrict our analysis to the portfolio
processes $\pi$, which guarantees that a unique solution to (1) exists, i.e.
to $\pi$ in

$\displaystyle\Lambda=\left\\{\pi=\big{(}\pi(t)\big{)}_{t\in[0,T]}\
\text{progr. measurable}\ \Big{|}\
\int_{0}^{T}z(t)\big{(}\pi(t)\big{)}^{2}dt<\infty\ Q-a.s.\right\\}$ (2)

If $\pi\in\Lambda$, it is straightforward to show that the unique solution
$V^{v_{0},\pi}(t)$ to (2) can be expressed in closed-form as

$\displaystyle V^{v_{0},\pi}(t)=v_{0}\exp\Big{(}\int_{0}^{t}r+\eta
z(s)\pi(s)-\frac{1}{2}z(s)\pi(s)^{2}ds+\int_{0}^{t}\pi(s)\sqrt{z(s)}dW(s)\Big{)}.\
$ (3)

For a closed convex set
$K\subset\mathbbm{R}\cup\\{\infty,-\infty\\}=:\bar{\mathbbm{R}}$ with non-
empty interior and CRRA utility function $U(v)=\frac{1}{b}v^{b}$ with $b<1$
and $b\neq 0,$ we consider the portfolio optimisation problem

$\mathbf{(P)}\begin{cases}\Phi(v_{0})&=\underset{\pi\in\Lambda_{K}}{\sup}\mathbbm{E}\big{[}U(V^{v_{0},\pi}(T))\big{]}\\\
\Lambda_{K}&=\left\\{\pi\in\Lambda\ \big{|}\ \pi(t)\in K\
\mathcal{L}[0,T]\otimes Q-\text{a.e.}\right\\}.\end{cases}$

As the considered financial market contains only one risky asset, the set of
allocation constraints $K$ is a subset of the extended real numbers
$\bar{\mathbbm{R}}$. However, in this one-dimensional setting, any such closed
convex set $K\subset\bar{\mathbbm{R}}$ with non-empty interior can be
expressed as an interval of the form222As any $\pi\in\Lambda$ can only take
finite values $\mathcal{L}[0,T]\otimes Q$-a.s., we do not need to distinguish
between $(-\infty,\beta]$ and $[-\infty,\beta]$ or $[\alpha,\infty)$ and
$[\alpha,\infty]$ for any $-\infty\leq\alpha,\beta\leq\infty$.

$\displaystyle
K=[\alpha,\beta],\quad\text{with}\quad-\infty\leq\alpha<\beta\leq\infty.$ (4)

We make substantial use of this in the subsequent analysis.

###  2.1 Solution to HJB PDE

We approach $(\mathbf{P})$ using classic stochastic optimal control methods.
For this purpose, let us introduce the generalised primal portfolio
optimisation problem $\mathbf{(P^{(t,v,z)})}$ as

$\mathbf{(P^{(t,v,z)})}\begin{cases}\Phi(t,v,z)&=\underset{\pi\in\Lambda_{K}(t)}{\sup}\mathbbm{E}\big{[}U(V^{v_{0},\pi}(T))\
\big{|}\ V^{v_{0},\pi}(t)=v,\ z(t)=z\big{]}\\\
\Lambda_{K}(t)&=\left\\{\left(\pi(s)\right)_{s\in[t,T]}\ \big{|}\
\pi\in\Lambda_{K}\right\\}.\end{cases}$

Then, the HJB equation associated with $\mathbf{(P^{(t,v,z)})}$ is given by

$\displaystyle 0$ $\displaystyle=\sup_{\pi\in K}\Big{(}$ $\displaystyle
G_{t}+v(r+\eta
z\pi)G_{v}+\kappa(\theta-z)G_{z}+\frac{1}{2}v^{2}z\pi^{2}G_{vv}+\rho
vz\pi\sigma G_{vz}+\frac{1}{2}\sigma^{2}zG_{zz}\Big{)}$ (5) $\displaystyle
G(T,v,z)$ $\displaystyle=U(v),$ (6)

Any (sufficiently regular) solution $G$ to (5) naturally yields a candidate
optimal portfolio $\pi^{\ast}$ to $\mathbf{(P^{(t,v,z)})}$ (and therefore
$\mathbf{(P)}$) as the maximising argument of (5). In their work on dynamic
stochastic-factor models, Escobar-Anel et al. (2023) characterised $G$ as an
exponentially affine function, whose exponents satisfy certain Riccati ODEs.
To recall their results, we need to introduce the support function
$\delta_{K}:\mathbbm{R}\rightarrow\bar{\mathbbm{R}}$ of $K$ as

$\displaystyle\delta_{K}(x)=-\inf_{y\in K}\left(x\cdot
y\right)\overset{K=[\alpha,\beta]}{=}-\alpha x\mathbbm{1}_{\\{x>0\\}}-\beta
x\mathbbm{1}_{\\{x<0\\}}.$

######  Lemma 2.1.

Let $A$ and $B$ be solutions to the system of ODEs

$\displaystyle A^{\prime}(\tau)$ $\displaystyle=br+\kappa\theta B(\tau),$ (7)
$\displaystyle B^{\prime}(\tau)$ $\displaystyle=-\kappa
B(\tau)+\frac{1}{2}\sigma^{2}\left(B(\tau)\right)^{2}+\frac{1}{2}\frac{b}{1-b}\inf_{\lambda\in\mathbbm{R}}\left(2(1-b)\delta_{K}(\lambda)+\left(\eta+\lambda+\sigma\rho
B(\tau)\right)^{2}\right),$ (8)

with initial condition $A(0)=B(0)=0.$ Then,
$G(t,v,z)=\frac{1}{b}v^{b}\exp\left(A(T-t)+B(T-t)z\right)$ is a solution to
(5).

Given the minimising argument and the solution $B$ from (8), a candidate
optimal portfolio $\pi^{\ast}$ for $(\mathbf{P})$ is known. If we define the
sequence of stopping times $\tau_{n,t}$ as
$\tau_{n,t}=\min(T,\hat{\tau}_{n,t})$, with

$\displaystyle\hat{\tau}_{n,t}=\inf\Big{\\{}t\leq u\leq T\ \Big{|}\ $
$\displaystyle\int_{t}^{u}\left(b\cdot\sqrt{z(s)}\cdot\pi(s)\cdot
G(s,V^{v_{0},\pi^{\ast}}(s),z(s))\right)^{2}ds\geq n,$
$\displaystyle\int_{t}^{u}\left(\sigma\sqrt{z(s)}\cdot B(T-s)\cdot
G(s,V^{v_{0},\pi^{\ast}}(s),z(s))\right)^{2}ds\geq n\Big{\\}},$

then we can give a uniform integrability condition which guarantees that the
candidate optimal portfolio $\pi^{\ast}$ is indeed optimal for $(\mathbf{P}).$

######  Lemma 2.2.

Consider $A,B$ and $G$ from Lemma 2.1. Moreover, define

$\displaystyle\lambda^{\ast}(B)=\underset{\lambda\in\mathbbm{R}}{\text{argmin}}\left\\{2(1-b)\delta_{K}(\lambda)+\left(\eta+\lambda+\sigma\rho
B\right)^{2}\right\\}$ (9) $\displaystyle\pi^{\ast}(t)$
$\displaystyle=\frac{1}{1-b}\Big{(}\eta+\lambda^{\ast}(B(T-t))+\sigma\rho
B(T-t)\Big{)}.$ (10)

If
$\left(G\left(\tau_{n,t},V^{v_{0},\pi^{\ast}}(\tau_{n,t}),z(\tau_{n,t})\right)\right)_{n\in\mathbbm{N}}$
is uniformly integrable for every $t\in[0,T],$ then $\pi^{\ast}$ is optimal
for $\mathbf{(P)}$ and $\Phi(t,z,v)=G(t,v,z)$ for all
$(t,v,z)\in[0,T]\times(0,\infty)\times(0,\infty).$

Lemma 2.1 and Lemma 2.2 naturally lead to a three-step procedure for finding
the optimal portfolio $\pi^{\ast}$ for $\mathbf{(P)}$:

  * (i)

Determine the minimising argument $\lambda^{\ast}$ in (9).

  * (ii)

Determine the solution $B$ to ODE (8) and thereby the candidate optimal
portfolio $\pi^{\ast}$ from (10).

  * (iii)

Verify that $\pi^{\ast}$ satisfies the uniform integrability condition from
Lemma 2.2.

In the following, we complete these steps consecutively and complete the
results of Escobar-Anel et al. (2023) by providing a fully closed-form
solution for the optimal allocation-constrained portfolio in Heston’s
stochastic volatility model via steps (i) and (ii) and formally verifying its
optimality in step (iii).

As $K$ is an interval, as specified in (4), the ODE (8) can be written as a
composition of three Riccati ODEs - each with constant coefficients.

######  Lemma 2.3.

Define $B_{-}=\frac{(1-b)\alpha-\eta}{\sigma}$ and
$B_{+}=\frac{(1-b)\beta-\eta}{\sigma}$. Then, the minimising argument
$\lambda^{\ast},$ as in (9), is given as

$\displaystyle\lambda^{\ast}(B)=$
$\displaystyle\big{[}(1-b)\alpha-\left(\eta+\sigma\rho
B\right)\big{]}\mathbbm{1}_{\\{\rho
B<B_{-}\\}}+\big{[}(1-b)\beta-\left(\eta+\sigma\rho
B\right)\big{]}\mathbbm{1}_{\\{\rho B>B_{+}\\}}.$ (11)

Moreover, $B(\tau)$ is a solution to $(\ref{eq: ODE B constr. Heston})$ if and
only if $B(0)=0$ and

$\displaystyle B^{\prime}(\tau)=\quad$
$\displaystyle\Big{(}-\underbrace{\frac{1}{2}b\alpha\big{(}(1-b)\alpha-2\eta\big{)}}_{=:r_{0}^{-}}+\underbrace{\big{(}b\sigma\rho\alpha-\kappa\big{)}}_{=:r_{1}^{-}}B(\tau)+\frac{1}{2}\underbrace{\sigma^{2}}_{=:r_{2}^{-}}\big{(}B(\tau)\big{)}^{2}\Big{)}\mathbbm{1}_{\\{\rho
B(\tau)<B_{-}\\}}$ $\displaystyle\quad+$
$\displaystyle\Big{(}-\underbrace{\frac{-b}{2(1-b)}\eta^{2}}_{=:r_{0}}+\underbrace{\big{(}\frac{b}{1-b}\eta\sigma\rho-\kappa\big{)}}_{=:r_{1}}B(\tau)+\frac{1}{2}\underbrace{\sigma^{2}\big{(}1+\frac{b}{1-b}\rho^{2}\big{)}}_{=:r_{2}}\big{(}B(\tau)\big{)}^{2}\Big{)}\mathbbm{1}_{\\{B_{-}\leq\rho
B(\tau)\leq B_{+}\\}}$ $\displaystyle\quad+$
$\displaystyle\Big{(}-\underbrace{\frac{1}{2}b\beta\big{(}(1-b)\beta-2\eta\big{)}}_{=:r_{0}^{+}}+\underbrace{\big{(}b\sigma\rho\beta-\kappa\big{)}}_{=:r_{1}^{+}}B(\tau)+\frac{1}{2}\underbrace{\sigma^{2}}_{=:r_{2}^{+}}\big{(}B(\tau)\big{)}^{2}\Big{)}\mathbbm{1}_{\\{B_{+}<\rho
B(\tau)\\}}$ (12) $\displaystyle=\quad$
$\displaystyle\Big{(}-r_{0}^{-}+r_{1}^{-}B(\tau)+\frac{1}{2}r_{2}^{-}+\big{(}B(\tau)\big{)}^{2}\Big{)}\mathbbm{1}_{\\{\rho
B(\tau)<B_{-}\\}}$ $\displaystyle\quad+$
$\displaystyle\Big{(}-r_{0}+r_{1}B(\tau)+\frac{1}{2}r_{2}\big{(}B(\tau)\big{)}^{2}\Big{)}\mathbbm{1}_{\\{B_{-}\leq\rho
B(\tau)\leq B_{+}\\}}$ $\displaystyle\quad+$
$\displaystyle\Big{(}-r_{0}^{+}+r_{1}^{+}B(\tau)+\frac{1}{2}r_{2}^{+}\big{(}B(\tau)\big{)}^{2}\Big{)}\mathbbm{1}_{\\{B_{+}<\rho
B(\tau)\\}}.$

######  Remark 1.

By restricting the minimisation in ODE (8) from $\lambda\in\mathbbm{R}$ to one
of the three optimal values $\lambda\in\\{(1-b)\alpha-\left(\eta+\sigma\rho
B\right),(1-b)\beta-\left(\eta+\sigma\rho B\right),0\\}$ (cf. (11)), we may
use (2.3) to write

$\displaystyle B^{\prime}(\tau)=-\kappa
B(\tau)+\frac{1}{2}\sigma^{2}\left(B(\tau)\right)^{2}+\frac{1}{2}\frac{b}{1-b}\inf_{\lambda\in\mathbbm{R}}\left(2(1-b)\delta_{K}(\lambda)+\left(\eta+\lambda+\sigma\rho
B(\tau)\right)^{2}\right)$
$\displaystyle=\min\left(-r_{0}^{-}+r_{1}^{-}B(\tau)+\frac{1}{2}r_{2}^{-}\big{(}B(\tau)\big{)}^{2},-r_{0}+r_{1}B(\tau)+\frac{1}{2}r_{2}\big{(}B(\tau)\big{)}^{2},-r_{0}^{+}+r_{1}^{+}B(\tau)+\frac{1}{2}r_{2}^{+}\big{(}B(\tau)\big{)}^{2}\right)$
$\displaystyle=:f(B(\tau)).$

The coefficients $r_{2}^{-},$ $r_{2}$ and $r_{2}^{+}$ are non-negative, and
therefore $f$ is the minimum of three convex functions. As real convex
functions are locally Lipschitz continuous and Lipschitz continuity is
preserved when taking the minimum over a finite number of functions, $f$ is
locally Lipschitz continuous too. Hence, by the existence and uniqueness
theorem of Picard-Lindelöf, there exists a unique solution $B$ to (8) for
small $\tau>0.$ Moreover, as $f$ does not depend on $\tau,$ the ODE for $B$ is
autonomous and its solution $B$ is either constant (if $f(0)=0$) or strictly
monotone in $\tau$ (if $f(0)\neq 0$). Analogous arguments can be used to
conclude the (strict) monotonicity of $B_{u}(\tau)$ from Corollary 2.10 in
Section 2.3.

######  Remark 2.

Note that if $B$ is a solution to (8), then Lemma 2.2 and Lemma 2.3 imply

$\displaystyle\pi^{\ast}(t)=\frac{\eta+\lambda^{\ast}(B(T-t))+\sigma\rho
B(T-t)}{1-b}=\begin{cases}\alpha,&\quad\rho B(T-t)<B_{-}\\\
\frac{\eta+\sigma\rho B(T-t)}{1-b},&\quad B_{-}\leq\rho B(T-t)\leq B_{+}\\\
\beta,&\quad B_{+}<\rho B(T-t).\\\ \end{cases}$

Therefore, the zones $Z_{-}=(-\infty,B_{-})$, $Z_{0}=[B_{-},B_{+}]$ and
$Z_{+}=(B_{+},\infty)$ determine whether the allocation constraint
$K=[\alpha,\beta]$ is enforced for the candidate optimal portfolio process
$\pi^{\ast}$ from Lemma 2.2. Moreover, we may define
$\hat{\pi}^{\ast}(t):=\frac{1}{1-b}\left(\eta+\sigma\rho B(T-t)\right)$ and
express $\pi^{\ast}$ as a capped version of $\hat{\pi}^{\ast}$, i.e.

$\pi^{\ast}(t)=\text{Cap}(\hat{\pi}^{\ast}(t),\alpha,\beta):=\begin{cases}\alpha,&\quad\hat{\pi}^{\ast}(t)<\alpha\\\
\hat{\pi}^{\ast}(t),&\quad\alpha\leq\hat{\pi}^{\ast}(t)\leq\beta\\\
\beta,&\quad\beta<\hat{\pi}^{\ast}(t).\end{cases}$

In a true constrained context, i.e. $K\neq\mathbbm{R}$, we may either
determine an approximation of $B$ by using a suitable numerical ODE solver
(such as an Euler method) to solve the ODE (2.3) or directly derive an
explicit expression for $B$ by individually solving each of the three Riccati
ODEs in (2.3) and merging the solutions at the transition points between the
zones $Z_{-}$, $Z_{0}$ and $Z_{+}$. To ensure that such solutions exist and do
not explode before time $T$, we need to make the following assumption on
$\mathcal{M}_{H}$ and the constraints $K=[\alpha,\beta].$

######  Assumption 2.4.

1

  * (i)

Existence of Solution:

$\displaystyle\max\left\\{\begin{array}[]{lr}\frac{b}{1-b}\eta\Big{(}\frac{\kappa\rho}{\sigma}+\frac{\eta}{2}\Big{)},\\\
b\alpha\left(\eta-\frac{1}{2}\alpha+\frac{\kappa\rho}{\sigma}+\frac{1}{2}\alpha
b(1-\rho^{2})\right),\\\
b\beta\left(\eta-\frac{1}{2}\beta+\frac{\kappa\rho}{\sigma}+\frac{1}{2}\beta
b(1-\rho^{2})\right)\end{array}\right\\}<\frac{\kappa^{2}}{2\sigma^{2}}$

  * (ii)

No Blow-Up:
The coefficients of each of the three Riccati ODEs satisfy $t_{+}(B_{0})>T$
(cf. Lemma B.4, (ii)) for each initial value

$B_{0}\in\left\\{\left(\frac{B_{-}}{\rho}\right)\mathbbm{1}_{\\{\rho\neq
0\\}},\left(\frac{B_{+}}{\rho}\right)\mathbbm{1}_{\\{\rho\neq
0\\}},0\right\\}.$

Provided that Assumption 2.4 holds, the coefficients

$\displaystyle r_{3}^{-}=\sqrt{(r_{1}^{-})^{2}+2r_{0}^{-}r_{2}^{-}},\quad
r_{3}=\sqrt{(r_{1})^{2}+2r_{0}r_{2}},\quad
r_{3}^{+}=\sqrt{(r_{1}^{+})^{2}+2r_{0}^{+}r_{2}^{+}}$ (13)

are well-defined and the solutions to each of the Riccati ODEs (2.3) do not
blow up before time $T$ when started at any of the transition points between
the zones $Z_{-}$, $Z_{0}$ and $Z_{+}$.333Technically, one can formulate this
assumption less restrictively by expressing ‘No Blow-Up’ in terms of the time
spent in each of the zones $Z_{-}$, $Z_{0}$ and $Z_{+}$. However, as this
would significantly complicate the presentation without adding major
additional insights, it is omitted here. For this reason, we define the
following auxiliary functions:

  * •

Let $\hat{B}^{+},$ $\hat{B},$ and $\hat{B}^{-}$ be the solution to Riccati ODE
(37) with initial value $0$ as well as coefficients $r_{0}^{+},$ $r_{1}^{+},$
$r_{2}^{+}$, $r_{0},$ $r_{1},$ $r_{2}$ and $r_{0}^{-},$ $r_{1}^{-},$
$r_{2}^{-},$ respectively.

  * •

If $\rho\neq 0$, let $\hat{B}_{+}^{+},$ $\hat{B}_{-}^{-}$ be the solution to
Riccati ODE (37) with initial value $\frac{B_{+}}{\rho}$, $\frac{B_{-}}{\rho}$
and coefficients $r_{0}^{+},$ $r_{1}^{+},$ $r_{2}^{+}$, $r_{0}^{-},$
$r_{1}^{-},$ $r_{2}^{-},$ respectively.

  * •

If $\rho\neq 0$, let $\hat{B}_{+},$ $\hat{B}_{-}$ be the solution to Riccati
ODE (37) with initial value $\frac{B_{+}}{\rho}$, $\frac{B_{-}}{\rho}$,
respectively, and coefficients $r_{0},$ $r_{1},$ $r_{2}$.

Moreover, if $\rho\neq 0$, we define the transition times444If $\rho=0$ all of
these transition times will be infinite.

$\displaystyle\tau_{1}^{+}$ $\displaystyle=\inf\left\\{\tau\ |\
\hat{B}^{+}(\tau)=\frac{B^{+}}{\rho}\right\\},\quad\tau_{2}^{+}=\inf\left\\{\tau\
|\
\hat{B}_{+}(\tau)=\frac{B^{-}}{\rho}\right\\},\quad\tau_{1}=\inf\left\\{\tau\
|\
\hat{B}(\tau)\in\left\\{\frac{B^{-}}{\rho},\frac{B^{+}}{\rho}\right\\}\right\\}$
$\displaystyle\tau_{1}^{-}$ $\displaystyle=\inf\left\\{\tau\ |\
\hat{B}^{-}(\tau)=\frac{B^{-}}{\rho}\right\\}\quad\text{and}\quad\tau_{2}^{-}=\inf\left\\{\tau\
|\ \hat{B}_{-}(\tau)=\frac{B^{+}}{\rho}\right\\}.$

Note that each of the above functions and transition times, if finite, admit a
closed-form expression, which can be obtained via Lemma B.4 and Corollary B.5
in the supplementary material. Having introduced these auxiliary functions and
transition times, we can finally express a closed-form solution for $B$ in
terms of these processes.

######  Theorem 2.5.

Let Assumption 2.4 hold. Then,

$\displaystyle
B(\tau)=\begin{cases}\hat{B}^{-}(\tau)\mathbbm{1}_{\\{\tau\leq\tau^{-}_{1}\\}}+\hat{B}_{-}(\tau-\tau^{-}_{1})\mathbbm{1}_{\\{\tau^{-}_{1}<\tau\leq\tau^{-}_{1}+\tau^{-}_{2}\\}}+\hat{B}_{+}^{+}(\tau-(\tau^{-}_{1}+\tau^{-}_{2}))\mathbbm{1}_{\\{\tau^{-}_{1}+\tau^{-}_{2}<\tau\\}},&\
\text{if}\ 0\in Z_{-}\\\
\hat{B}(\tau)\mathbbm{1}_{\\{\tau\leq\tau_{1}\\}}+\hat{B}_{-}^{-}(\tau-\tau_{1})\mathbbm{1}_{\\{\tau>\tau_{1},\rho\hat{B}(\tau_{1})=B_{-}\\}}+\hat{B}_{+}^{+}(\tau-\tau_{1})\mathbbm{1}_{\\{\tau>\tau_{1},\rho\hat{B}(\tau_{1})=B_{+}\\}},&\
\text{if}\ 0\in Z_{0}\\\
\hat{B}^{+}(\tau)\mathbbm{1}_{\\{\tau\leq\tau^{+}_{1}\\}}+\hat{B}_{+}(\tau-\tau^{+}_{1})\mathbbm{1}_{\\{\tau^{+}_{1}<\tau\leq\tau^{+}_{1}+\tau^{+}_{2}\\}}+\hat{B}_{-}^{-}(\tau-(\tau^{+}_{1}+\tau^{+}_{2}))\mathbbm{1}_{\\{\tau^{+}_{1}+\tau^{+}_{2}<\tau\\}},&\
\text{if}\ 0\in Z_{+}\\\ \end{cases}$

satisfies ODE (8) for $0\leq\tau\leq T.$555Using a similar separation with
respect to the zones $Z_{-},$ $Z_{0},$ and $Z_{+}$ and equation (39), it is
also possible to determine a closed-form expression for $A$ from Lemma 2.1.

###  2.2 Verification Theorem

Combining Remark 2 with Theorem 2.5 immediately yields a closed-form
expression for the candidate optimal portfolio process $\pi^{\ast}$. It now
just remains to prove a verification theorem which verifies that this
candidate is indeed the optimal portfolio process corresponding to
$\mathbf{(P)}$. This proof requires an additional assumption on the
constraints $K=[\alpha,\beta],$ which ensures a certain boundedness of
$\pi^{\ast}(t)$ for $t$ close to maturity $T$ as well as two auxiliary lemmas.

######  Assumption 2.6.

$\displaystyle\max\left\\{\frac{b\rho}{\kappa}\alpha,\
\frac{b\rho}{\kappa}\beta\right\\}\leq\frac{\kappa}{\sigma^{2}},$ (14)

######  Lemma 2.7.

Let Assumptions 2.4 and 2.6 hold and let $B$ be given as in Theorem 2.5. Then,
the following inequality holds for all $t\in[0,T]$

$\displaystyle\frac{b\rho}{\sigma}\pi^{\ast}(t)+B(T-t)\leq\frac{\kappa}{\sigma^{2}}.$

######  Lemma 2.8.

Let Assumptions 2.4 and 2.6 hold and let $B$ be given as in Theorem 2.5. Then
the following inequality holds for all $t\in[0,T]$

$\displaystyle\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-t)\right)+\sigma\rho
B(T-t)\right)^{2}-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(s)\right)^{2}$
$\displaystyle+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(t)+\frac{b}{1-b}\frac{\rho}{\sigma}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-t)\right)+\sigma\rho\right]B^{\prime}(T-t)\quad$
$\displaystyle<\frac{1}{2}\frac{\kappa^{2}}{\sigma^{2}}.$

######  Theorem 2.9 (Verification Theorem in $\mathcal{M}_{H}$).

1
Consider the financial market $\mathcal{M}_{H}$, let Assumptions 2.4 and 2.6
hold and let $B$ be given as in Theorem 2.5. Then,

$\displaystyle\pi^{\ast}(t)=\begin{cases}\alpha,&\quad\rho B(T-t)<B_{-}\\\
\frac{\eta+\sigma\rho B(T-t)}{1-b},&\quad B_{-}\leq\rho B(T-t)\leq B_{+}\\\
\beta,&\quad B_{+}<\rho B(T-t).\\\ \end{cases}$ (15)

is optimal for $\mathbf{(P)}$.

###  2.3 Comparison to Unconstrained Portfolio

Unsurprisingly, we can immediately recover the solution to the unconstrained
optimisation problem, as discussed in Kraft (2005), from Lemma 2.2 and Lemma
2.3.

######  Corollary 2.10.

[Closed-form Unconstrained Optimal Portfolio as in Kraft (2005)] 1
Let $K=\mathbbm{R}$ (i.e. $\alpha=-\infty,$ $\beta=\infty)$ and
$B_{u}:[0,T]\rightarrow\mathbbm{R}$ with $B_{u}(0)=0$ satisfy

$\displaystyle
B_{u}^{\prime}(\tau)=-r_{0}+r_{1}B_{u}(\tau)+\frac{1}{2}r_{2}B_{u}(\tau)^{2}\qquad\forall\tau\in[0,T].$
(16)

Then, $\lambda^{\ast}(B)=0$ $\forall B\in\mathbbm{R}$ and the candidate
optimal portfolio $\pi^{\ast}$ is given by

$\pi_{u}(t):=\pi^{\ast}(t)=\frac{1}{1-b}\left(\eta+\sigma\rho
B_{u}(T-t)\right).$

######  Remark 3.

If the market parameters satisfy (cf. Assumption 2.4)

$\displaystyle\frac{b}{1-b}\eta\Big{(}\frac{\kappa\rho}{\sigma}+\frac{\eta}{2}\Big{)}<\frac{\kappa^{2}}{2\sigma^{2}},$
(17)

then

$\displaystyle
B_{u}(\tau)=\frac{2r_{0}(e^{r_{3}\tau}-1)}{(r_{1}-r_{3})(e^{r_{3}\tau}-1)-2r_{3}}$
(18)

and the optimality of $\pi_{u}$ for the unconstrained portfolio optimisation
problem can be verified formally (see e.g. Theorem 5.3 in Kraft
(2005)).666Equation (17) corresponds to part (i) of Assumption 2.4. In the
setting of Kraft (2005), part (ii) of Assumption 2.4 is also implied by (17)
and so does not have to be mentioned explicitly.

On an abstract level, when adding (allocation) constraints $K=[\alpha,\beta]$
to a portfolio optimisation problem, the optimal constrained portfolio
$\pi^{\ast}$ for $(\mathbf{P})$ will be given by a projection
$\mathcal{P}_{K}:\Lambda\rightarrow\Lambda_{K}$ which maps the optimal
unconstrained portfolio $\pi_{u}$ onto $\Lambda_{K},$ i.e.

$\pi^{\ast}=\pi_{u}+\left(\pi^{\ast}-\pi_{u}\right)=:\mathcal{P}_{K}\left(\pi_{u}\right).$

In a Black-Scholes financial market $\mathcal{M}_{BS}$ with constant market
coefficients (i.e. $\mathcal{M}_{H}$ with $\sigma=\kappa=\rho=0.$), the
optimal unconstrained portfolio is a constant-mix strategy
$\pi_{u}(t):=\pi_{M}=\frac{1}{1-b}\eta,$ the so-called ‘Merton portfolio’.
Setting $\sigma=\rho=0$ and $B\equiv 0$ in Remark 2, one can easily see that
the projection $\mathcal{P}=\mathcal{P}^{BS}$ in the Black-Scholes market
simply caps off $\pi_{M}$ at the boundaries if $\pi_{M}\notin
K=[\alpha,\beta],$ i.e.

$\mathcal{P}^{BS}_{K}(\pi_{M})=\text{Cap}\left(\pi_{M},\alpha,\beta\right)=\begin{cases}\alpha,&\quad\pi_{M}<\alpha\\\
\pi_{M},&\quad\alpha\leq\pi_{M}\leq\beta\\\
\beta,&\quad\beta<\pi_{M}.\end{cases}$

Given a solution $B$ to (8) and considering Remark 2, it initially appears
that the optimal constrained portfolio $\pi^{\ast}$ in $\mathcal{M}_{H}$ can
be obtained from the same projection. However, if $K\neq\mathbbm{R},$ then
$B_{u}$ as in Corollary 2.10 and $B$ as in Theorem 2.5 are solutions to
possibly different ODEs. In particular, this implies that the portfolios
$\pi_{u}$ and $\hat{\pi}^{\ast}$ may not be identical, in which case the
projection $\mathcal{P}^{H}_{K}$ for the Heston market does not necessarily
coincide with the projection $\mathcal{P}^{BS}_{K}$ for the Black-Scholes
market either. In other words, in a financial market with Heston stochastic
volatility we in general have

$\displaystyle\pi^{\ast}=\mathcal{P}^{H}_{K}\left(\pi_{u}\right)=\text{Cap}\big{(}\pi_{u}+\underbrace{\left(\hat{\pi}^{\ast}-\pi_{u}\right)}_{\neq
0},\alpha,\beta\big{)}\neq\text{Cap}\big{(}\pi_{u},\alpha,\beta\big{)}=\mathcal{P}^{BS}_{K}(\pi_{u}).$

In the following, we render this observation more precise by providing both
conditions under which $\mathcal{P}^{H}_{K}=\mathcal{P}^{BS}_{K}$ and
conditions under which $\mathcal{P}^{H}_{K}\neq\mathcal{P}^{BS}_{K}.$ The
former case is true, whenever either $\rho=0$ or $\pi_{M}\in K.$

######  Lemma 2.11.

Let $\pi^{\ast}$ be as in Lemma 2.2, $\hat{\pi}^{\ast}$ be as in Remark 2 and
$\pi_{u}$ as in Corollary 2.10. If either

$\rho=0\qquad\text{or}\qquad\pi_{M}\in K,$

then

$\pi^{\ast}=\mathcal{P}^{H}_{K}\left(\pi_{u}\right)=\text{Cap}\left(\pi_{u},\alpha,\beta\right)=\mathcal{P}^{BS}_{K}\left(\pi_{u}\right).$

If $\rho=0$, the stochasticity of the volatility is completely unhedgeable in
$\mathcal{M}_{H}$. As a consequence, the optimal unconstrained portfolio
processes coincide in $\mathcal{M}_{BS}$ and in $\mathcal{M}_{H}.$ Thus, the
projections $\mathcal{P}^{BS}_{K}$ and $\mathcal{P}^{H}_{K}$ are identical if
$\rho=0$ too. In contrast, if $\rho\neq 0,$ then the projections can only be
different if the underlying ODE solutions $B_{u}$ and $B$ are different,
specifically when $\pi_{u}$ and $\hat{\pi}^{\ast}$ begin taking values inside
$K.$ This is the case if and only if $\pi_{u}$ and $\hat{\pi}^{\ast}$ begin
taking values inside $K$ at different time points. This observation leads to
an equivalent characterisation of when the projections $\mathcal{P}^{BS}_{K}$
and $\mathcal{P}^{H}_{K}$ are different.

######  Lemma 2.12.

Let $\pi^{\ast}$ be as in Lemma 2.2, $\hat{\pi}^{\ast}$ be as in Remark 2 and
$\pi_{u}$ as in Corollary 2.10. The following statements are equivalent:

  * (i)

$\pi^{\ast}=\mathcal{P}^{H}_{K}\left(\pi_{u},\alpha,\beta\right)\neq\mathcal{P}^{BS}_{K}\left(\pi_{u},\alpha,\beta\right)=\text{Cap}(\pi_{u},\alpha,\beta)$

  * (ii)

$\pi_{M}\notin[\alpha,\beta]\quad\text{and}\quad\exists
t\in(0,T):\left|\Big{\\{}\hat{\pi}^{\ast}(t),\pi_{u}(t)\Big{\\}}\cap(\alpha,\beta)\right|=1$

We can construct an extreme case which satisfies the requirements of Lemma
2.12 by choosing $\alpha$ such that $B(\tau)$ is constant and choosing the
market parameters such that $\pi_{u}$ changes sufficiently during the
investment horizon to ensure that $\pi_{u}(t^{\ast})\in(\alpha,\beta)$ for
some $t^{\ast}\in[0,T].$

######  Corollary 2.13.

Let $\pi^{\ast}$ be as in Lemma 2.2, $\hat{\pi}^{\ast}$ be as in Remark 2 and
$\pi_{u}$ as in Corollary 2.10. Let $\text{sign}(x)\in\\{-1,0,1\\}$ denote the
sign of $x\in\mathbbm{R}.$

  * (i)

If

$0<\pi_{M}=\frac{\alpha}{2}<\alpha\quad\text{and}\quad\alpha<\pi_{u}(t^{\ast})<\beta\quad\text{for
some}\ t^{\ast}\in[0,T],$

then $B(\tau)=\alpha$ for all $\tau\in[0,T],$ $\pi^{\ast}(t)=\alpha$ for all
$t\in[0,T]$ and
$\pi^{\ast}=\mathcal{P}^{H}_{K}\left(\pi_{u},\alpha,\beta\right)\neq\mathcal{P}^{BS}_{K}\left(\pi_{u},\alpha,\beta\right)=\text{Cap}\left(\pi_{u},\alpha,\beta\right).$

  * (ii)

If $\pi_{M}>\beta>0,$ then

$\text{sign}\left(\frac{\partial}{\partial
t}\hat{\pi}^{\ast}(t)\right)=\text{sign}\left(\frac{\partial}{\partial
t}\pi_{u}(t)\right)=-\text{sign}(\rho b)\quad\forall t\in[0,T].$

Hence, if in addition $b<0$ and $\rho<0,$ then
$\mathcal{P}^{H}_{K}=\mathcal{P}^{BS}_{K}.$

Clearly, the requirements on $\alpha$ in Corollary 2.13, (i) are quite
restrictive, but they still provide a valuable insight into when we can expect
to see a large difference between the projections $\mathcal{P}^{BS}_{K}$ and
$\mathcal{P}^{H}_{K}.$ Namely, if

  * •

the optimal unconstrained portfolio $\pi_{u}$ violates the constraint at
maturity (i.e. $\pi_{u}(T)=\pi_{M}\notin K$) and there is sufficient change in
$\pi_{u}(t)$ during the investment period such that $\pi_{u}(t^{\ast})\in K$
for some $t^{\ast}\in K.$

  * •

the derivatives of $B(\tau)$ and $B_{u}(\tau)$ are considerably different
while $\pi_{u}\notin K$ (constant $B$ being the extreme case).

As a matter of fact, we will later see in the numerical experiments in Section
4 that it is sufficient if $\alpha\approx 2\pi_{M}$ (i.e. $B(\tau)$ is nearly
constant) to cause a considerable difference between the two projections.

As evidenced by the majority of empirical calibrations of Heston’s stochastic
volatility model to financial time series (see e.g. Escobar-Anel and
Gschnaidtner (2016) for an overview), the parameter $\rho$ is negative for
most realistic applications. In the empirical study on risk preferences of
mutual fund managers by Koijen (2014), it is reported that the risk aversion
parameter $b$ has a median of $b=-1.43$ and a mean of $b=-4.8.$ For more risk
averse investors, such as insurance companies, reinsurance companies or
pension funds, one can thus realistically assume negative values for $b.$
Thus, for most realistic parameter configurations of $\mathcal{M}_{H}$ with
$\pi_{M}>\beta,$ the projections $\mathcal{P}^{H}_{K}$ and
$\mathcal{P}^{BS}_{K}$ coincide for investors with a high degree of risk
aversion (i.e. for a low value of $b$).

##  3 Implications for Related Models

We now consider a generalised version of the financial market
$\mathcal{M}_{H}$ with $d\in N$ risky assets, $d$ independent CIR processes as
risk drivers and a generalised dependence of market price of risk and risky
asset volatility on these risk drivers. From now on, let $W^{z}$ and $\hat{W}$
denote independent $d$-dimensional Wiener processes and consider parameters
$\kappa,$ $\theta,$ $\sigma,$ $z_{0}\in(0,\infty)^{d}$ such that their
components satisfy $2\kappa_{i}\theta_{i}>\sigma_{i}^{2}$ for $i=1,...,d.$
Then, we define the $d-$dimensional CIR process
$z=\left(z_{1},...,z_{d}\right)^{\prime}$ through the dynamics

$\displaystyle
dz_{i}(t)=\kappa\left(\theta_{i}-z_{i}(t)\right)dt+\sigma_{i}\sqrt{z_{i}(t)}dW^{z}_{i}(t).$

In the following, let $\mathbbm{1}\in\mathbbm{R}^{d}$ be the $d$-dimensional
vector of ones, let $x\odot y$ denote the element-wise multiplication of
$x,y\in\mathbbm{R}^{d}$ and let $\sqrt{x}$ denote the element-wise square root
of $x\in\mathbbm{R}^{d}.$ Consider a given correlation vector
$\rho\in(-1,1)^{d},$ market price of risk
$\gamma:(0,\infty)^{d}\rightarrow\mathbbm{R}^{d}$ and volatility
$\Sigma:(0,\infty)^{d}\rightarrow\mathbbm{R}^{d\times d}$ such that
$\Sigma(z)$ is non-singular for all $z\in(0,\infty)^{d}.$ Then, we define the
financial market $\mathcal{M}_{H}^{\gamma,\Sigma},$ consisting of one risk-
free asset $P_{0}$ and $d$ risky assets
$P=\left(P_{1},\ldots,P_{d}\right)^{\prime}$ with dynamics

$\displaystyle dP_{0}(t)$ $\displaystyle=P_{0}(t)rdt$ $\displaystyle dP(t)$
$\displaystyle=P(t)\odot\big{[}\underbrace{\left(r\mathbbm{1}+\Sigma(z(t))\gamma(z(t))\right)}_{=:\mu(z(t))}dt+\Sigma(z(t))\underbrace{\left(\rho\cdot
dW^{z}(t)+\sqrt{\mathbbm{1}-\rho\odot\rho}d\hat{W}(t)\right)}_{=:dW(t)}\big{]}.$

Clearly, we can recover the financial market $\mathcal{M}_{H},$ as considered
in Section 2, by assuming $d=1$ and choosing $\gamma(z)=\eta\sqrt{z}$ and
$\Sigma(z)=\sqrt{z}$ and the Black Scholes model $\mathcal{M}_{BS}$ if both
$\gamma$ and $\Sigma$ are constants. Similar, but slightly more general
financial market models than $\mathcal{M}_{H}^{\gamma,\Sigma}$ have been
considered in Liu (2006) or Escobar-Anel et al. (2023), for example. In
$\mathcal{M}_{H}^{\gamma,\Sigma},$ the wealth process $V^{v_{0},\pi}$ of an
investor with initial wealth $v_{0}$ who trades continuously in time with
$\mathbbm{R}^{d}$-valued relative portfolio process $\pi$, satisfies the SDE

$\displaystyle dV^{v_{0},\pi}(t)$
$\displaystyle=V^{v_{0},\pi}(t)\left[\left(r+\gamma(z(t))^{\prime}\Sigma(z(t))^{\prime}\pi(t)\right)dt+\pi(t)^{\prime}\Sigma(z(t))dW(t)\right]$
$\displaystyle=V^{v_{0},\pi}(t)\left[\left(r+\left(\mu(z(t))-r\mathbbm{1}\right)^{\prime}\pi(t)\right)dt+\pi(t)^{\prime}\Sigma(z(t))dW(t)\right].$

In $\mathcal{M}_{H}^{\gamma,\Sigma},$ the set of admissible portfolio
processes naturally generalises to

$\displaystyle\Lambda^{\gamma,\Sigma}=\left\\{\pi=\big{(}\pi(t)\big{)}_{t\in[0,T]}\
\text{progr. measurable}\ \Big{|}\
\int_{0}^{T}\|\Sigma(z(t))^{\prime}\pi(t)\|^{2}dt<\infty\ Q-a.s.\right\\}$

For a closed convex set with non-empty interior
$K\subset\bar{\mathbbm{R}}^{d},$ the portfolio optimisation problem
$\mathbf{(P)}$ in $\mathcal{M}_{H}^{\gamma,\Sigma}$ is then defined as

$\mathbf{(P)}\begin{cases}\Phi(v_{0})&=\underset{\pi\in\Lambda_{K}}{\sup}\mathbbm{E}\big{[}U(V^{v_{0},\pi}(T))\big{]}\\\
\Lambda_{K}&=\big{\\{}\pi\in\Lambda^{\gamma,\Sigma}\ \big{|}\ \pi(t)\in K\
\mathcal{L}[0,T]\otimes Q-\text{a.e.}\big{\\}}.\end{cases}$

In the following two sections, we investigate the solvability of
$\mathbf{(P)}$ for given choices of $\gamma,$ $\Sigma,$ and $K.$ In Section
3.1, we consider the PCSV Model, as discussed in Escobar-Anel et al. (2017),
and in Section 3.2, we consider inverse volatility constraints $K,$ which
impose stronger restrictions on an investor’s portfolio during periods of high
volatility.

###  3.1 PCSV Model

We recover the PCSV (‘Principal Component Stochastic Volatility’) model
$\mathcal{M}^{PCSV},$ as proposed in Escobar-Anel et al. (2010), from
$\mathcal{M}_{H}^{\gamma,\Sigma}$ by considering an orthogonal matrix
$A=\big{(}a_{1},...,a_{d}\big{)}\in\mathbbm{R}^{d\times d}$ and defining
market price of risk and volatility as

$\displaystyle\gamma(z)=\text{diag}(\sqrt{z})A^{\prime}\eta=\Sigma(z)^{\prime}\eta,\quad\Sigma(z)=A\text{diag}(\sqrt{z})\quad\forall
z\in(0,\infty)^{d},$ (19)

where $\text{diag}(x)\in\mathbbm{R}^{d\times d}$ denotes the diagonal matrix
with entries $x\in\mathbbm{R}^{d},$ and $\eta\in\mathbbm{R}^{d}$ is a
constant.
If $A=I$, then $\mathcal{M}^{PCSV}$ can be regarded as a canonical
generalisation of Heston’s stochastic volatility model $\mathcal{M}_{H}$ for a
$d$-dimensional asset universe, where each risky asset’s volatility is
determined as the square root of one of the $d$ independent CIR processes
$z_{i}.$ However, in its general form, the independent components of the
$d$-dimensional CIR process $z$ are not directly regarded as volatilities.
Instead, the instantaneous covariance matrix of risky asset returns

$\displaystyle\Sigma(z(t))\Sigma(z(t))^{\prime}=A\text{diag}(z(t))A^{\prime}$

is decomposed into its principal components, i.e. the columns $a_{i}$ of the
matrix $A$ represent its eigenvectors, and the independent CIR processes
$z_{i}$ represent their (stochastic) eigenvalues. This approach not only
enables the modelling of stochastic covariances of asset returns because of
additional degrees of freedom in $A,$ but also allows for an interpretation of
$z$ as hidden risk factors determining the volatility level in the financial
market. Moreover, Escobar-Anel et al. (2017) demonstrated that several
stylised facts are captured by the PCSV, such as stochasticity of volatilities
and correlation of risky asset returns, volatility and correlation leverage
effect, volatility spillovers, and increasing correlation in periods of high
market volatility.
Let $\Lambda^{PCSV}$ be the set of admissible portfolios in
$\mathcal{M}^{PCSV}.$ Then, for any $\pi\in\Lambda^{PCSV},$ the wealth process
$V^{v_{0},\pi}$ satisfies the SDE

$\displaystyle
dV^{v_{0},\pi}(t)=V^{v_{0},\pi}(t)\left[\left(r+\eta^{\prime}A\text{diag}(z(t))A^{\prime}\pi(t)\right)dt+\pi(t)^{\prime}A\text{diag}(\sqrt{z(t)})dW(t)\right],$

for $t\in[0,T].$ The instantaneous variance of $V^{v_{0},\pi}(t)$ can
therefore be decomposed into a weighted sum of the risk factors $z,$ since

$\displaystyle\|\text{diag}(\sqrt{z(t)})A^{\prime}\pi(t)\|=\left\|\begin{pmatrix}a_{1}^{\prime}\pi(t)\sqrt{z_{1}(t)}\\\
\vdots\\\
a_{d}^{\prime}\pi(t)\sqrt{z_{d}(t)}\end{pmatrix}\right\|^{2}=\sum_{i=1}^{d}\left(a_{i}^{\prime}\pi(t)\right)^{2}z_{i}(t).$

In this sense, the portfolio weights determine a risk exposure
$\left(a_{i}^{\prime}\pi(t)\right)^{2}$ to the risk factor $z_{i}.$ Hence, it
is very natural to impose risk limits on these exposures, i.e. for given upper
bounds $\beta_{1},...,\beta_{d}>0$ we require that

$\displaystyle\left(a_{i}^{\prime}\pi(t)\right)^{2}\leq\beta_{i}\quad\forall
i=1,...,d\quad\Leftrightarrow\quad
A^{\prime}\pi(t)\in\bigtimes_{i=1}^{d}[0,\sqrt{\beta_{i}}]\quad\Leftrightarrow\quad\pi(t)\in\underbrace{A\cdot\left(\bigtimes_{i=1}^{d}[0,\sqrt{\beta_{i}}]\right)}_{=:K_{PCSV}}.$

We can reuse the ideas and results from Section 2 to obtain the optimal
portfolio to the portfolio optimisation problem $\mathbf{(P)}$ in
$\mathcal{M}_{H}^{\gamma,\Sigma}=\mathcal{M}_{PCSV}$ with constraints
$K=K_{PCSV}.$

######  Theorem 3.1.

Consider the portfolio optimisation problem $\mathbf{(P)}$ in
$\mathcal{M}_{PCSV}$ with constraints $K=K_{PCSV}$, let the parameters $b,$
$\left(\eta_{A}\right)_{i}:=\left(A^{\prime}\eta\right)_{i}$, $\kappa_{i},$
$\theta_{i},$ $\sigma_{i},$ $\alpha_{i}=0,$ $\sqrt{\beta_{i}}$ satisfy
Assumptions 2.4 and 2.6 and $B_{i}$ be defined as in Theorem 2.5. Define the
portfolio $\pi^{\ast}_{A}(t)$ via

$\displaystyle\left(\pi^{\ast}_{A}(t)\right)_{i}=\text{Cap}\left(\frac{1}{1-b}\left(\eta_{i}+\sigma\rho
B_{i}(T-t)\right),0,\sqrt{\beta_{i}}\right).$

Then the portfolio $\pi^{\ast}(t)=A\cdot\pi^{\ast}_{A}(t)$ is optimal for
$\mathbf{(P)}.$

The key argument in the proof of Theorem 3.1 lies in a change of control,
which transforms the portfolio optimisation problem $\mathbf{(P)}$ into an
equivalent portfolio optimisation problem $\mathbf{(P_{A})}$ in a financial
market, which consists of $d$ risky assets with independent Heston
volatilities and interval constraints. Thanks to this familiar structure and
the independence of the risky asset volatilities, we can extend the ideas from
Section 2 to solve $\mathbf{(P_{A})}$ and invert the change of control to
obtain a solution to $\mathbf{(P)}.$

###  3.2 Inverse Volatility Constraints

We now discuss a related problem, which we approach with a similar methodology
to the one in Section 3.1. Consider again the one-dimensional setting with one
risky asset, i.e. $d=1.$ In this section, we no longer assume that the convex
constraints $K\subset\bar{\mathbbm{R}}$ are static, but allow them to depend
on the stochastic factor $z.$ More specifically, we consider volatility-
dependent constraints of the form

$\displaystyle\pi(t)\in K(z(t))\quad\mathcal{L}[0,T]\otimes Q-a.e.,$

where $K:(0,\infty)\rightarrow\mathcal{B}(\mathbbm{R})$ is a set-valued
function, taking only closed-convex values in the Borel set
$\mathcal{B}(\mathbbm{R}).$ The motivation for such constraints is quite
clear: Depending on the current state of the financial market
$\mathcal{M}_{H}^{\gamma,\Sigma},$ in particular the level of risky asset
volatility $\Sigma(z(t))$ and the market price of risk $\gamma(z(t)),$
investors may face different constraints on their portfolio, such as more
relaxed bounds in periods of low volatility or stricter bounds in periods of
high volatility. Further, in the spirit of mean-variance optimisation, we can
think of an investor seeking an optimal portfolio allocation subject to
constraints on his instantaneous portfolio volatility

$\displaystyle
0\leq\pi(t)\Sigma(z(t))\leq\beta_{z}\quad\mathcal{L}[0,T]\otimes
Q-a.e.\quad\Leftrightarrow\quad
0\leq\pi(t)\leq\frac{\beta_{z}}{\Sigma(z(t))}\quad\mathcal{L}[0,T]\otimes
Q-a.e.,$

for a given volatility level $\beta_{z}>0.$777Note that this is different from
classic mean-variance optimisation, where the variance of the terminal
portfolio wealth $V^{v_{0},\pi}(T)$ is constrained.

Keeping this motivation in mind, we thus define the portfolio optimisation
problem with volatility-dependent constraints $\mathbf{(P^{z})}$ as

$\displaystyle\mathbf{(P^{z})}=\begin{cases}\Phi^{z}(v_{0})&=\underset{\pi\in\Lambda_{K(\cdot)}}{\sup}\mathbbm{E}\big{[}U(V^{v_{0},\pi}(T))\big{]}\\\
\Lambda_{K(\cdot)}&=\left\\{\pi\in\Lambda^{\gamma,\Sigma}\ \big{|}\ \pi(t)\in
K(z(t))\ \mathcal{L}[0,T]\otimes Q-\text{a.e.}\right\\}.\end{cases}$

In its most general form, the portfolio optimisation $\mathbf{(P^{z})}$ is
highly non-trivial, since closed-form solutions for its optimal portfolio
process $\pi^{\ast}_{z}$ can rarely be determined for general $\gamma$ and
$\Sigma,$ even in the absence of (stochastic) allocation constraints. In
particular, the portfolio optimisation $\mathbf{(P)}$ is included as a special
case in the definition of $\mathbf{(P^{z}).}$ However, due to the results of
Cvitanic and Karatzas (1992) for the Black-Scholes model $\mathcal{M}^{BS}$
with constant volatility, as well as the results from Section 2 for Heston’s
stochastic volatility model $\mathcal{M}^{H},$ we know of at least two
different models in which solutions to $\mathbf{(P)}$ (respectively
$\mathbf{(P^{z})}$) with static constraints can be obtained in closed form.
Using another change of control argument, we can therefore derive conditions
on the market parameters $\gamma,$ $\Sigma$ and the constraints $K,$ under
which we can transform $\mathbf{(P^{z})}$ into an equivalent, solvable
portfolio optimisation problem $\mathbf{(P)}$ in either $\mathcal{M}^{BS}$ or
$\mathcal{M}^{H}.$

######  Theorem 3.2.

Consider the financial market $\mathcal{M}_{H}^{\gamma,\Sigma}$ and the
portfolio optimisation problem $\mathbf{(P^{z})}.$ Consider constants
$-\infty\leq\alpha<\beta\leq\infty.$

  * (i)

If $\gamma(z)=\eta$ for some $\eta>0$ and
$K(z)=\frac{1}{\Sigma(z)}[\alpha_{z},\beta_{z}],$ then the portfolio process

$\displaystyle\pi^{\ast}_{z}(t)=\frac{1}{\Sigma\left(z(t)\right)}\text{Cap}(\pi_{M},\alpha,\beta)$

is optimal for $\mathbf{(P^{z})}.$

  * (ii)

If $\gamma(z)=\eta\sqrt{z(t)}$ for some constant $\eta>0,$
$K(z)=\frac{\sqrt{z}}{\Sigma(z)}[\alpha,\beta],$ and Assumptions 2.4 and 2.6
are satisfied, then for $\pi^{\ast}$ defined as in Theorem 2.9, the portfolio
process

$\displaystyle\pi^{\ast}_{z}(t)=\frac{\sqrt{z(t)}}{\Sigma\left(z(t)\right)}\pi^{\ast}(t)$

is optimal for $\mathbf{(P^{z})}.$

The statements of Theorem 3.2 can easily be easily generalised to financial
markets with $d>1$ risky assets by an analogous change of control argument.
Using the results for constant volatility markets from Example 15.2 in
Cvitanic and Karatzas (1992), one can prove a multi-dimensional analogue to
statement (i) and using the results for the PCSV model from Section 3.1, one
can prove a multi-dimensional analogue to statement (ii). For ease of
presentation, we refrain from a detailed discussion of this generalisation.

##  4 Numerical Studies

In this section, we illustrate the properties of the optimal portfolio
$\pi^{\ast}$ for $\mathbf{(P)}$ in Heston’s stochastic volatility model
$\mathcal{M}_{H},$ using a numerical example. In particular, we analyse the
difference between $\pi^{\ast}$ and two suboptimal naive portfolio processes
$\pi,$ which either directly follow the optimal portfolio process in
$\mathcal{M}_{BS}$ (i.e. $\pi=\text{Cap}(\pi_{M},\alpha,\beta)$) or apply the
projection $\mathcal{P}^{BS}_{K}$ from $\mathcal{M}_{BS}$ to the optimal
unconstrained portfolio $\pi_{u}$ in $\mathcal{M}_{H}$ (i.e.
$\pi=\text{Cap}(\pi_{u},\alpha,\beta)$). The suboptimality of these portfolios
will be quantified using the concept of wealth-equivalent loss.

Such an analysis is only meaningful if the differences between a financial
market with stochastic (Heston) volatility $\mathcal{M}_{H}$ and a financial
market with constant volatility $\mathcal{M}_{BS}$ are already reflected in
the optimal unconstrained portfolios $\pi_{u}$ for $\mathcal{M}_{H}$ and
$\pi_{M}$ for $\mathcal{M}_{BS}.$ Since allocation constraints further
restrict the set of admissible portfolio allocations, any existing differences
between $\pi_{u}$ and $\pi_{M}$ tend to be diminished further when adding
allocation constraints. From an investor’s perspective, the distinction
between $\mathcal{M}_{BS}$ and $\mathcal{M}_{H}$ is only relevant if the
volatility of risky asset log returns $\sqrt{z(t)}$ changes significantly and
these changes are partially hedgeable through trading in the risky asset. This
is the case if the volatility of the volatility ($\sigma$) is large, the mean
reversion speed ($\kappa$) is small, and the correlation between risky asset
and volatility diffusion ($\rho$) is close to either $1$ or $-1$ (i.e.
$|\rho|$ is large).
Based on these requirements, we choose the market parameters (see Table 1) for
our numerical example such that the resulting market dynamics resemble a
financial crisis.

Parameter |  | Value | Explanation
---|---|---|---
End of Investment-Horizon | $T$ | $1$ | Limited duration of financial crises
Risk Aversion Parameter | $b$ | $-2.5$ | Within ranges estimated in Table 1, Koijen (2014)
Initial Wealth | $v_{0}$ | $1$ | For convenience
Risk-Free Interest Rate | $r$ | $0$ | For convenience
Market Price of Risk Driver | $\eta$ | $3.0071$ | Table 2, Cheng and Escobar-Anel (2021)
Mean Reversion Speed | $\kappa$ | $3.15$ | Table 3 ‘%MSE’, Moyaert and Petitjean (2011)
Volatility of Volatility | $\sigma$ | $0.76$ | Table 3 ‘%MSE’, Moyaert and Petitjean (2011)
Correlation | $\rho$ | $-0.81$ | Table 3 ‘%MSE’, Moyaert and Petitjean (2011)
Long-Term Mean | $\theta$ | $0.35$ | Feller’s Condition
Initial Variance | $z_{0}$ | $0.35$ | Chosen equal to $\theta$
Table 1: Base parameters for the financial market $\mathcal{M}_{H}.$

The only volatility parameters which influence the optimal portfolio
allocation are $\sigma,$ $\kappa,$ and $\rho.$ Our choices for these
parameters in Table 1, follow the calibration results of Moyaert and Petitjean
(2011), who calibrated Heston’s stochastic volatility model using option
prices on the Eurostoxx 50 during the 2008 financial crisis. The relatively
short investment horizon of $T=1$ year is chosen to reflect the limited
duration of most financial crises.888Q.ai (2022) reported that the average
length of an S&P500 bear market (defined as a period with drawdown in excess
of $20\%$) was $289$ days.
We quantify the sub-optimality of both naive portfolio processes in comparison
to the optimal constrained portfolio process using the concept of wealth-
equivalent loss (‘WEL’). For an arbitrary portfolio process
$\pi\in\Lambda_{K},$ we define the expected utility functional
$J^{\pi}:[0,T]\times(0,\infty)\times(0,\infty)\rightarrow\mathbbm{R}$ as

$\displaystyle J^{\pi}(t,v,z)=\mathbbm{E}\left[U\left(V^{v_{0},\pi}(T)\right)\
|\ V^{v_{0},\pi}(t)=v,\ z(t)=z\right].$ (20)

When considering the optimal portfolio process $\pi^{\ast},$ the expected
utility functional coincides with the value function of $\mathbf{(P)},$ i.e.
$J^{\pi^{\ast}}(t,v,z)=\Phi(t,v,z)$ for all
$(t,v,z)\in[0,T]\times(0,\infty)\times(0,\infty).$ The WEL
$L^{\pi}=L^{\pi}(t,z)$ of $\pi$ is then defined as the solution to the
equation999Since we exclusively work with power utility functions in this
paper, we may without loss of generality assume that the WEL is independent of
wealth.

$\displaystyle\Phi(t,v(1-L^{\pi}(t,z)),z)=J^{\pi}(t,v,z).$

An investor following the optimal portfolio allocation $\pi^{\ast}$ only needs
$(1-L^{\pi}(t,z))$ as much capital to achieve the same average utility as an
investor following the sub-optimal strategy $\pi.$ In this sense,
$L^{\pi}(t,z)$ can be interpreted as a relative loss incurred for investing
sub-optimally.101010If $\pi$ is deterministic and $J^{\pi}$ is the unique
solution to the Feynman-Kac PDE, one can use an exponentially affine ansatz to
characterise $J^{\pi}$ in terms of the solutions to a system of ODEs. If the
ODE solutions are given, then the WEL $L^{\pi}(0,z_{0})$ is known in closed
form. We provide a description of this approach in Lemma B.6 and Corollary B.7
in the supplementary material. In our studies, we approximated the
corresponding ODE solutions by an Euler method.

###  4.1 Optimal Constrained Merton Portfolio
$\pi=\text{Cap}\left(\pi_{M},\alpha,\beta\right)$

In this subsection, we compare $\pi^{\ast}$ with the first naive portfolio
process $\pi=\text{Cap}(\pi_{M},\alpha,\beta).$ Although $\pi$ is static, it
is at least known that $\pi$ is optimal for an allocation constrained
portfolio optimisation problem in the Black-Scholes market $\mathcal{M}_{BS},$
whereas no theoretical guarantees were available for the corresponding
optimisation problem in $\mathcal{M}_{H}$ prior to this work. During this
analysis, we consider the allocation constraint $K=[\alpha,\beta]=[0,1],$
which corresponds to a no-borrowing constraint that prevents short-selling in
the risk-free and risky asset.

Figure 1: Portfolio weights $\pi(t)$ for $t\in[0,T]$, lower bound $\alpha=0,$
upper bound $\beta=1,$ and parameters as in Table 1.

Assuming the parameters in Table 1, the Merton portfolio satisfies the
constraints, i.e. $\pi_{M}\in[\alpha,\beta]$ and thus
$\text{Cap}(\pi_{M},\alpha,\beta)=\pi_{M}.$ In contrast to this constant
allocation in the interior of $[\alpha,\beta],$ $\pi^{\ast}$ initially takes a
constant value at the upper bound $\beta$ and later decreases towards
$\pi_{M}$ at the end of the investment horizon. Therefore, other than
$\text{Cap}(\pi_{M},\alpha,\beta),$ $\pi^{\ast}$ is able to realise and
benefit from a higher allocation to the risky asset. Note that
$\pi_{M}\in[\alpha,\beta]$ implies
$\pi^{\ast}=\text{Cap}(\pi_{u},\alpha,\beta),$ as shown in Lemma 2.11.

In the following, we quantify the impact of the suboptimal allocations
$\pi=\text{Cap}(\pi_{M},\alpha,\beta)$ by computing the annual WEL
$L^{\pi}(0,z)$ at the beginning of the investment horizon and analyse its
sensitivity with respect to the risk-aversion parameter $b$ and the volatility
drivers $\sigma,$ $\kappa$ and $\rho.$ The ranges of the volatility parameter
are chosen to be within the minimum and maximum parameter values obtained in
individual calibrations in Table 5 of Moyaert and Petitjean (2011).

Figure 2: Dependence of the annual wealth-equivalent loss $L^{\pi}(0,z_{0})$
on individual parameters for $\pi=\text{Cap}(\pi_{M},\alpha,\beta)=\pi_{M},$
lower bound $\alpha=0,$ upper bound $\beta=1,$ and parameters as in Table 1.
Figure 2 illustrates the dependence on $b\in[-10,0.7]\backslash\\{0\\},$
whereas Figures 2, 2 and 2 display the dependence on $\sigma\in[0.2,1.0],$
$\kappa\in[1.5,5.0]$ and $\rho\in[-0.9,-0.4],$ respectively.

For small values of $b,$ where the allocation constraint $K=[0,1]$ is largely
satisfied by the unconstrained portfolios $\pi_{M}$ and $\pi_{u},$ the WELs
displayed in Figure 2 are increasing in $b.$ However, as $b$ increases past an
inflection point of approximately $b=-3,$ the allocation constraint $K$
becomes active. From this point onwards, $K$ forces $\pi^{\ast}$ and
$\text{Cap}(\pi_{M},\alpha,\beta)$ closer towards each other for increasing
$b$ and therefore leads to decreasing WELs. Ultimately, for $b\geq-2,$ we have
$\pi^{\ast}(t)=\text{Cap}(\pi_{M},\alpha,\beta)=\beta=1$ for all $t\in[0,T]$
and thus the WEL is zero. Figures 2, 2 and 2 display WELs which are increasing
in $\sigma$ as well as decreasing in $\kappa$ and $\rho.$ This confirms the
intuition voiced at the beginning of Section 4, in which we argued that
$\mathcal{M}_{H}$ is ‘more different’ to $\mathcal{M}_{BS}$ under these
circumstances. Within the chosen parameter ranges, we observe the largest
annual WEL of $3.2\%$ for small $\kappa,$ whereas increasing $\sigma$ and
decreasing $\rho$ leads to WELs of $3.0\%$ and $2.5\%,$ respectively. Changing
any one of the values of the volatility parameters $\sigma,$ $\kappa$ or
$\rho$ to less extreme levels, which are obtained during calibrations on long-
term data sets111111For an overview, consider e.g. Table 4 in Escobar-Anel and
Gschnaidtner (2016). Here, the authors consider values of $\kappa=3.5,$
$\sigma=0.3,$ $\rho=-0.4$ as an ‘Average Case’ for their reviewed literature.
leads to significant decreases in annual WELs. Even if only $\sigma<0.5,$
while $\kappa$ and $\rho$ remain at crisis level, the annual WEL still drops
to values of $0.75\%$ or lower. Note that Feller’s condition is satisfied for
all parameter values that were considered in our analysis.

###  4.2 Capped Optimal Unconstrained Heston Portfolio
$\pi=\text{Cap}\left(\pi_{u},\alpha,\beta\right)$

In this subsection, we compare $\pi^{\ast}$ to the second naive portfolio
process, the capped optimal unconstrained Heston portfolio
$\text{Cap}\left(\pi_{u},\alpha,\beta\right).$ In particular, we would like to
illustrate the phenomenon described in Section 2.3. Despite having a
theoretical guarantee that $\text{Cap}\left(\pi_{u},\alpha,\beta\right)$ is
indeed different from $\pi^{\ast}$ for certain parameter settings, these
differences appear to be mostly meaningful in terms of WEL for extreme market
scenarios in combination with specific, large lower bounds $\alpha.$ According
to Lemma 2.12 and Corollary 2.13, we know that
$\text{Cap}\left(\pi_{u},\alpha,\beta\right)$ and $\pi^{\ast}$ are identical
in the parameter setting of Table 1, unless $\pi_{M}<\alpha.$ Further,
Corollary 2.13 suggests that we should consider market parameters which ensure
$\pi_{u}(t)>2\pi_{M}$ for some $t\in[0,T].$
For these reasons, we adjust the previously considered parameter setting. In
the following, we choose the most extreme volatility parameters from the
sensitivity analysis in Figure 2, i.e. we set $\sigma=1.0,$ $\kappa=1.5$ and
$\rho=-0.9,$ and increase the risk aversion coefficient to $b=-15$ to obtain
realistic portfolio allocations.

Figure 3: Portfolio weights $\pi(t)$ for $t\in[0,T],$ upper bound $\beta=1,$
and parameters as in Table 1, except for $b=15,$ $\sigma=1.0,$ $\kappa=1.5$
and $\rho=-0.9.$ Figure 3 considers a lower bound $\alpha=1.75\pi_{M},$ and
Figure 3 considers a lower bound $\alpha=2\pi_{M}.$

Figure 3 compares the portfolio weights of $\pi^{\ast}(t)$ and
$\text{Cap}(\pi_{u}(t),\alpha,\beta)$ for lower bounds
$\alpha\in\\{1.75\pi_{M},2\pi_{M}\\}$ such that the corresponding ODE solution
$B$ is (nearly) constant, as described in Corollary 2.13. If
$\alpha=1.75\pi_{M}$ then $\pi^{\ast}$ is initially larger than the lower
bound $\alpha,$ then decreases until $\alpha$ is reached, while $\pi^{\ast}$
is constant for $\alpha=2\pi_{M}.$ Additionally, we observe that

$\pi^{\ast}(t)\leq\text{Cap}(\pi_{u}(t),\alpha,\beta)\quad\forall t\in[0,T],$

with equality only if $\pi_{u}(t)\leq\alpha.$ In both cases illustrated in
Figure 3, $\pi^{\ast}$ lowers the portfolio allocation early throughout the
investment horizon, thus accounting for the fact that the lower bound forces
the portfolio allocation to be larger than the optimal unconstrained
allocation later during the investment horizon.

Figure 4: Lower bounds
$\alpha\in\\{1.5\pi_{M},1.75\pi_{M},1.9\pi_{M},1.95\pi_{M},2\pi_{M}\\},$ upper
bound $\beta=1,$ $t\in[0,T],$ and parameters as in Table 1, except for $b=15,$
$\sigma=1.0,$ $\kappa=1.5$ and $\rho=-0.9.$ Figure 4 displays the portfolio
weights $\pi^{\ast}(t),$ and Figure 4 displays the WEL $\L^{\pi}(t,z_{0})$ for
$\pi=\text{Cap}\left(\pi_{u},\alpha,\beta\right).$

Figure 4 illustrates the behaviour of $\pi^{\ast}$ for varying $\alpha.$ When
increasing $\alpha,$ we observe that $\pi^{\ast}$ decreases to the lower bound
at earlier time points $t.$ Despite $\pi^{\ast}$ being constant for
$\alpha=1.95\pi_{M},$ the solution $B$ to the ODE (2.3) for $B$ is not
stationary, but $\rho B(\tau)$ does not leave zone $Z_{-}$ for $\tau\leq T.$
Thus, we expect $\pi^{\ast}$ to be constant for all
$\alpha\in[1.95\pi_{M},2\pi_{M}]$ in our parameter setting. Figure 4 displays
WELs $L^{\pi}(t,z_{0})$ of $\pi=\text{Cap}(\pi_{u},\alpha,\beta),$ which are
increasing in $\alpha$ at $t=0.$ However, this monotonicity does not hold
throughout the entire investment horizon, as increasing the lower bound
$\alpha$ implies that $\pi^{\ast}$ and $\text{Cap}(\pi_{u},\alpha,\beta)$
coincide for longer parts of the investment horizon.
Clearly, Figures 3 and 4 suggest a strong link between the value of $\alpha$
and the difference between $\pi^{\ast}$ and
$\text{Cap}(\pi_{u},\alpha,\beta).$ Therefore, we quantify this difference not
only using WELs, but additionally define the maximum absolute weight
difference between $\pi^{\ast}$ and a portfolio $\pi$ as

$\displaystyle\Delta^{\pi}_{\max}:=\max_{t\in[0,T]}\Big{|}\pi(t)-\pi^{\ast}(t)\Big{|}.$
(21)

The relationship between the lower bound $\alpha$ and the maximum absolute
difference $\Delta^{\pi}_{\max}$ and the annual WEL $L^{\pi}(0,z_{0})$ are
analysed in Figure 5.

Figure 5: Lower bounds $\alpha\in[\pi_{M},2\pi_{M}],$ upper bound $\beta=1,$
$\pi=\text{Cap}\left(\pi_{u},\alpha,\beta\right),$ and parameters as in Table
1, except for $b=15,$ $\sigma=1.0,$ $\kappa=1.5$ and $\rho=-0.9.$ For
$\pi=\text{Cap}\left(\pi_{u},\alpha,\beta\right),$ Figure 5 displays the
maximum absolute difference $\Delta^{\pi}$ to the portfolio weights
$\pi^{\ast}(t),$ and Figure 5 displays the WEL $\L^{\pi}(0,z_{0})$ to
$\pi^{\ast}.$

For $\pi=\text{Cap}(\pi_{u},\alpha,\beta),$ the maximum absolute difference
$\Delta^{\pi}_{\max}$ is generally increasing with $\alpha$ except for larger
lower bounds $\alpha.$ For large $\alpha,$ the optimal constrained portfolio
$\pi^{\ast}$ is constant throughout the investment horizon, as illustrated in
Figure 4. Then, the monotonicity of $\pi_{u}$ (see Remark 1) and
$\pi^{\ast}(T)=\alpha=\text{Cap}(\pi_{u}(T),\alpha,\beta)$ ensure that the
maximum in (21) is attained at $t=0.$ Further, $\pi_{u}(0)>2\pi_{M}\geq\alpha$
does not depend on $\alpha$. Therefore, if $\pi^{\ast}(t)=\alpha$ for all
$t\in[0,T],$ then the difference

$\Delta^{\text{Cap}(\pi_{u},\alpha,\beta)}_{\max}=\max_{t\in[0,T]}\Big{|}\text{Cap}(\pi_{u}(t),\alpha,\beta)-\pi^{\ast}(t)\Big{|}=\Big{|}\pi_{u}(0)-\alpha\Big{|}=\pi_{u}(0)-\alpha$

decreases linearly in $\alpha,$ which causes a slight kink in Figure 5 for
large lower bounds $\alpha.$ Irrespectively, the annual WEL for
$\pi=\text{Cap}(\pi_{u},\alpha,\beta)$ is increasing with $\alpha.$ However,
note that for all but very large lower bounds (e.g. $\alpha\geq 1.75\pi_{M}$),
the annual WEL is still negligible.

##  5 Conclusion

In this paper, we considered a portfolio optimisation problem with allocation
constraints in Heston’s stochastic volatility model. We derived an explicit
expression for the optimal portfolio and analysed its properties.
Surprisingly, this portfolio can be different from the naive constrained
portfolio which caps off the optimal unconstrained portfolio at the boundaries
of the constraint. In light of this fact, we have shown that the addition of
allocation constraints can have a fundamentally different impact on the
optimal portfolio in markets with stochastic volatility as compared to a
Black-Scholes market with constant volatility - even in financial markets with
only one risky asset. Irrespective of these theoretical certainties, we
observed in a numerical study that the annual wealth equivalent loss incurred
due to trading according to this naive portfolio is relatively small for the
majority of realistic scenarios. In this sense, the naive ‘capped’ portfolio
is nearly optimal for most applications. However, in turbulent financial
markets, such as the financial crisis of 2008, investors with a high degree of
risk aversion and lower bound on their portfolio allocation can suffer high
wealth equivalent losses. For such scenarios, investors should be mindful of
the optimal constrained portfolio and the naive capped portfolio.

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##  Appendix A Proofs

###### Proof of Lemma 2.1.

Follows immediately from Lemma 4.5 in Escobar-Anel et al. (2023) with $d=m=1$
and $\Sigma_{1}=1.$ ∎

###### Proof of Lemma 2.2.

Noting that $G_{v}(t,v,z)=\frac{b}{v}G(t,v,z)$ and
$G_{z}(t,v,z)=B(T-t)G(t,v,z)$, this result follows immediately from Theorem
3.12 and Lemma 4.5 in Escobar-Anel et al. (2023) with $d=m=1$ and
$\Sigma_{1}=1.$ ∎

###### Proof of Lemma 2.3.

The main objective of this proof is to determine the minimising argument
$\lambda^{\ast}(B)$ for the minimisation problem

$\inf_{\lambda\in\mathbbm{R}}\underbrace{\left(2(1-b)\delta_{K}(\lambda)+\left(\eta+\lambda+\sigma\rho
B\right)^{2}\right)}_{=:D(\lambda,B)}=\inf_{\lambda\in\mathbbm{R}}D(\lambda,B)$

for any fixed value $B\in\mathbbm{R}$. The remaining statement follows
immediately after substituting the obtained minimiser
$\lambda^{\ast}(B)=\lambda^{\ast}(B(\tau))$ in (8).

Given any constraints $K=[\alpha,\beta]$ with
$-\infty\leq\alpha\leq\beta\leq\infty$ (see (4)), we have for any
$\lambda\in\mathbbm{R}$

$\delta_{K}(\lambda)=-\inf_{\alpha\leq
x\leq\beta}(x\lambda)=-\alpha\lambda\mathbbm{1}_{\\{\lambda>0\\}}-\beta\lambda\mathbbm{1}_{\\{\lambda<0\\}}.$

Thus, taking the derivative of $D(\cdot,B)$ on $(-\infty,0)$ and $(0,\infty)$
yields

$\frac{\partial}{\partial\lambda}D(\lambda,B)=\begin{cases}-2(1-b)\alpha+2(\eta+\lambda+\sigma\rho
B),&\quad\lambda>0\\\ -2(1-b)\beta+2(\eta+\lambda+\sigma\rho
B),&\quad\lambda<0.\end{cases}$

Then, by the first-order optimality condition and the fact that $D(\cdot,B)$
is quadratic on $[0,\infty),$

$\displaystyle\lambda^{-}(B)$
$\displaystyle=\big{(}(1-b)\alpha-(\eta+\sigma\rho
B)\big{)}\mathbbm{1}_{\\{(1-b)\alpha-(\eta+\sigma\rho
B)>0\\}}=\big{(}(1-b)\alpha-(\eta+\sigma\rho B)\big{)}\mathbbm{1}_{\\{\rho
B<B_{-}\\}}$

minimises $D(\cdot,B)$ on $[0,\infty),$ and by the same argument

$\displaystyle\lambda^{+}(B)$
$\displaystyle=\big{(}(1-b)\beta-(\eta+\sigma\rho
B)\big{)}\mathbbm{1}_{\\{(1-b)\beta-(\eta+\sigma\rho
B)<0\\}}=\big{(}(1-b)\beta-(\eta+\sigma\rho B)\big{)}\mathbbm{1}_{\\{\rho
B>B_{+}\\}}$

minimises $D(\cdot,B)$ on $(-\infty,0]$. Therefore, noting that by
construction

$\displaystyle\max\big{(}D(\lambda^{-}(B),B),\ D(\lambda^{+}(B),B)\big{)}$
$\displaystyle=\max\Big{(}\underbrace{\min_{\lambda\in(-\infty,0]}D(\lambda,B)}_{\leq
D(0,B)},\underbrace{\min_{\lambda\in[0,\infty)}D(\lambda,B)}_{\leq
D(0,B)}\Big{)}$ $\displaystyle\quad\leq D(0,B)=(\eta+\sigma\rho B)^{2}$ (22)

and $B_{-}\leq B_{+}$, we finally derive

$\displaystyle\inf$ ${}_{\lambda\in\mathbbm{R}}D(\lambda,B)$
$\displaystyle=\min\big{(}D(\lambda^{-}(B),B),\ D(\lambda^{+}(B),B)\big{)}$
$\displaystyle=\min\Big{(}\big{[}2(1-b)(-\alpha)\big{(}(1-b)\alpha-\eta-\sigma\rho
B\big{)}+\big{(}(1-b)\alpha\big{)}^{2}\big{]}\mathbbm{1}_{\\{\rho
B<B_{-}\\}}+\big{[}\eta+\sigma\rho B\big{]}^{2}\mathbbm{1}_{\\{B_{-}\leq\rho
B\\}},$ $\displaystyle\hskip
42.67912pt\big{[}2(1-b)(-\beta)\big{(}(1-b)\beta-\eta-\sigma\rho
B\big{)}+\big{(}(1-b)\beta\big{)}^{2}\big{]}\mathbbm{1}_{\\{B_{+}<\rho
B\\}}+\big{[}\eta+\sigma\rho B\big{]}^{2}\mathbbm{1}_{\\{\rho B\leq
B_{+}\\}}\Big{)}$ $\displaystyle\overset{(\ref{eq: minimiser on both zones are
better than lambda =
0})}{=}\big{[}2(1-b)(-\alpha)\big{(}(1-b)\alpha-\eta-\sigma\rho
B\big{)}+\big{(}(1-b)\alpha\big{)}^{2}\big{]}\mathbbm{1}_{\\{B_{-}\leq\rho
B\\}}+\big{[}\eta+\sigma\rho B\big{]}^{2}\mathbbm{1}_{\\{B_{-}\leq\rho B\leq
B_{+}\\}}$
$\displaystyle\qquad\quad+\big{[}2(1-b)(-\beta)\big{(}(1-b)\beta-\eta-\sigma\rho
B\big{)}+\big{(}(1-b)\beta\big{)}^{2}\big{]}\mathbbm{1}_{\\{B_{+}<\rho B\\}}$
$\displaystyle=D(\lambda^{-}(B),B)\mathbbm{1}_{\\{B_{-}\leq\rho
B\\}}+D(0,B)\mathbbm{1}_{\\{B_{-}\leq\rho B\leq
B_{+}\\}}+D(\lambda^{+}(B),B)\mathbbm{1}_{\\{B_{+}<\rho B\\}}.$

Thus, the minimiser $\lambda^{\ast}(B)$ is given as

$\displaystyle\lambda^{\ast}(B)$
$\displaystyle=\lambda^{-}(B)\mathbbm{1}_{\\{\rho
B<B_{-}\\}}+\lambda^{+}(B)\mathbbm{1}_{\\{B_{+}<\rho B\\}}$
$\displaystyle=\big{(}(1-b)\alpha-(\eta+\sigma\rho
B)\big{)}\mathbbm{1}_{\\{\rho B<B_{-}\\}}+\big{(}(1-b)\beta-(\eta+\sigma\rho
B)\big{)}\lambda^{+}(B)\mathbbm{1}_{\\{B_{+}<\rho B\\}}.$

Substituting $\lambda^{\ast}(B(\tau))$ in (8) and factoring $B(\tau)$ and
$(B(\tau))^{2}$ concludes the proof. ∎

###### Proof of Theorem 2.5.

$B$ is specifically constructed in such a way that it is continuous and
satisfies the corresponding Riccati ODE of (2.3), whenever $\rho B(\tau)$ is
within each of the zones $Z_{-},$ $Z_{0},$ or $Z_{+}$. Moreover, as the right
hand side of the ODE (8) is continuous, so are the derivatives of the
constructed $B$. Thus, $B$ is a solution to (2.3) and thereby (8). ∎

###### Proof of Lemma 2.7.

By rewriting $\pi^{\ast}$ in terms of $\lambda^{\ast}$ and $B$, we see that

$\frac{b\rho}{\sigma}\pi^{\ast}(t)+B(T-t)=\begin{cases}\frac{b\rho}{\sigma}\alpha+B(T-t),&\
\text{if }\rho B(T-t)<B_{-}\\\
\frac{b}{1-b}\frac{\rho}{\sigma}\eta+\left(1+\frac{b}{1-b}\rho^{2}\right)B(T-t),&\
\text{if }B_{-}\leq\rho B(T-t)\leq B_{+}\\\
\frac{b\rho}{\sigma}\beta+B(T-t),&\ \text{if }B_{+}<\rho B(T-t)\\\
\end{cases}$

is non-decreasing in $B(T-t).$ Hence, as $B$ is itself monotonous (see Remark
1), we have

$\displaystyle\sup_{t\in[0,T]}\left(\frac{b\rho}{\sigma}\pi^{\ast}(t)+B(T-t)\right)=\max\left(\frac{b\rho}{\sigma}\pi^{\ast}(T)+B(0),\frac{b\rho}{\sigma}\pi^{\ast}(0)+B(T)\right).$
(23)

Hence, when first assuming that the maximum in (23) is attained at $t=T$, we
obtain

$\displaystyle\frac{b\rho}{\sigma}\underbrace{\pi^{\ast}(T)}_{\in[\alpha,\beta]}+\underbrace{B(0)}_{=0}\leq\max\left(\frac{b\rho}{\sigma}\alpha,\frac{b\rho}{\sigma}\beta\right)\overset{(\ref{eq:
boundedness assumption on constraints at
t=T})}{\leq}\frac{\kappa}{\sigma^{2}}.$

On the other hand, if we assume that the maximum in (23) is attained at $t=0$,
we need to compare the different terminal values $B(T)$ based on the
coefficients of the corresponding Riccati ODE at $T$. Due to the monotonicity
of $B$ and the assumption that the maximum in (23) is attained at $t=0$, it
suffices if we can find a bound for $T\rightarrow\infty$.

  * •

Case $\rho B(T)\in Z_{+}$:
Then, $\pi^{\ast}(0)=\beta$ and $B(T)$ is given as the solution (38) to the
Riccati ODE (37) with coefficients $r^{+}_{0},$ $r^{+}_{1},$ and $r^{+}_{2}$.
In particular, we have

$\displaystyle\frac{b\rho}{\sigma}$
$\displaystyle\underbrace{\pi^{\ast}(0)}_{=\beta}+B(T)\leq\frac{b\rho}{\sigma}\beta+\lim_{T\rightarrow\infty}B(T)$
$\displaystyle=\frac{b\rho}{\sigma}\beta+\lim_{T\rightarrow\infty}\left(\frac{2r^{+}_{2}r^{+}_{3}B_{0}+\left(e^{r^{+}_{3}T}-1\right)(r^{+}_{1}+r^{+}_{3})\left(r^{+}_{1}+r^{+}_{2}B_{0}-r^{+}_{3}\right)}{2r^{+}_{2}r^{+}_{3}-r^{+}_{2}\left(e^{r^{+}_{3}T}-1\right)\left(r^{+}_{1}+r^{+}_{2}B_{0}-r^{+}_{3}\right)}\right)$
$\displaystyle=\frac{b\rho}{\sigma}\beta-\frac{r^{+}_{1}+r^{+}_{3}}{r^{+}_{2}}=\frac{b\rho}{\sigma}\beta-\frac{b\sigma\rho\beta-\kappa}{\sigma^{2}}-\frac{r^{+}_{3}}{\sigma^{2}}=\frac{\kappa}{\sigma^{2}}-\underbrace{\frac{r^{+}_{3}}{\sigma^{2}}}_{\geq
0}\leq\frac{\kappa}{\sigma^{2}}.$

  * •

Case $\rho B(T)\in Z_{-}$:
Analogous arguments as in the case $\rho B(T)\in Z_{+}$ yield that

$\displaystyle\frac{b\rho}{\sigma}\pi^{\ast}(0)+B(T)\leq\frac{\kappa}{\sigma^{2}}.$

  * •

Case $\rho B(T)\in Z_{0}$: Then,
$\pi^{\ast}(0)=\frac{1}{1-b}\left(\eta+\sigma\rho B(T)\right)$ and $B(T)$ is
given as the solution (38) to Riccati ODE (37) with coefficients $r_{0},$
$r_{1},$ and $r_{2}$. In particular, we have

$\displaystyle\frac{b\rho}{\sigma}$
$\displaystyle\pi^{\ast}(0)+B(T)=\frac{\rho}{\sigma}\frac{b}{1-b}\left(\eta+\sigma\rho
B(T)\right)+B(T)=\frac{\rho}{\sigma}\frac{b}{1-b}\eta+\left(1+\frac{b}{1-b}\rho^{2}\right)B(T)$
$\displaystyle\leq\frac{\rho}{\sigma}\frac{b}{1-b}\eta+\left(1+\frac{b}{1-b}\rho^{2}\right)\underbrace{\lim_{T\rightarrow\infty}B(T)}_{=-\frac{r_{1}+r_{3}}{r_{2}}}=\frac{\rho}{\sigma}\frac{b}{1-b}\eta-\left(1+\frac{b}{1-b}\rho^{2}\right)\frac{r_{1}+r_{3}}{r_{2}}$
$\displaystyle=\frac{\rho}{\sigma}\frac{b}{1-b}\eta-\left(1+\frac{b}{1-b}\rho^{2}\right)\frac{r_{1}+r_{3}}{\sigma^{2}\left(1+\frac{b}{1-b}\rho^{2}\right)}=\frac{\rho}{\sigma}\frac{b}{1-b}\eta-\frac{r_{1}+r_{3}}{\sigma^{2}}$
$\displaystyle=\frac{\rho}{\sigma}\frac{b}{1-b}\eta-\frac{1}{\sigma^{2}}\left(\frac{b}{1-b}\eta\sigma\rho-\kappa\right)-\frac{r_{3}}{\sigma^{2}}=\frac{\kappa}{\sigma^{2}}-\underbrace{\frac{r_{3}}{\sigma^{2}}}_{\geq
0}\leq\frac{\kappa}{\sigma^{2}}.$

∎

###### Proof of Lemma 2.8.

Note that we may express ODE (8) in terms of $\lambda^{\ast}$ and $\pi^{\ast}$
as

$\displaystyle B^{\prime}(\tau)=-\kappa
B(\tau)+\frac{1}{2}\sigma^{2}B(\tau)^{2}+\frac{1}{2}\frac{b}{1-b}\left(2(1-b)\delta_{K}\left(\lambda^{\ast}\left(B(\tau)\right)\right)+(1-b)^{2}\left(\pi^{\ast}(T-\tau)\right)^{2}\right).$

We distinguish between three cases, depending on whether the allocation
constraint is active or not.

  * •

Case $\pi^{\ast}(t)\in(\alpha,\beta)$:
Then,
$\lambda^{\ast}\left(B(T-t)\right)=\left(\lambda^{\ast}\right)^{\prime}\left(B(T-t)\right)=\delta_{K}\left(\lambda^{\ast}\left(B(T-t)\right)\right)=0$.
Hence,

$\displaystyle\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-t)\right)+\sigma\rho
B(T-t)\right)^{2}-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(t)\right)^{2}$
$\displaystyle\quad+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(t)+\frac{b}{1-b}\frac{\rho}{\sigma}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-t)\right)+\sigma\rho\right]B^{\prime}(T-t)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\sigma\rho
B(T-t)\right)^{2}-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(t)\right)^{2}$
$\displaystyle\quad+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(t)+\frac{b}{1-b}\rho^{2}\left[-\kappa
B(T-t)+\frac{1}{2}\sigma^{2}\left(B(T-t)\right)^{2}+\frac{1}{2}b(1-b)\left(\pi^{\ast}(t)\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\sigma\rho
B(T-t)\right)^{2}+\frac{b}{1-b}\frac{\rho\kappa}{\sigma}\left(\eta+\sigma\rho
B(T-t)\right)$ $\displaystyle\quad+\rho^{2}\frac{b}{1-b}\left[-\kappa
B(T-t)+\frac{1}{2}\sigma^{2}\left(B(T-t)\right)^{2}\right]$ $\displaystyle=$
$\displaystyle\frac{b}{1-b}\eta\left(\frac{\eta}{2}+\frac{\rho\kappa}{\sigma}\right)$

  * •

Case $\pi^{\ast}(t)=\alpha$:
Then, $\lambda^{\ast}\left(B(T-t)\right)=(1-b)\alpha-\eta-\sigma\rho B(T-t)$
and

$\underbrace{\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-t)\right)+\sigma\rho\right]}_{=0}B^{\prime}(T-t)=0.$

Hence,

$\displaystyle\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\overbrace{\left(\lambda^{\ast}\left(B(T-t)\right)+\sigma\rho
B(T-t)\right)^{2}}^{=\left((1-b)\alpha-\eta\right)^{2}}-\frac{1}{2}b^{2}\rho^{2}\underbrace{\left(\pi^{\ast}(t)\right)^{2}}_{=\alpha^{2}}$
$\displaystyle\quad+b\frac{\rho\kappa}{\sigma}\underbrace{\pi^{\ast}(t)}_{=\alpha}+\frac{b}{1-b}\frac{\rho}{\sigma}\underbrace{\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-t)\right)+\sigma\rho\right]B^{\prime}(T-t)}_{=0}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left((1-b)\alpha-\eta\right)^{2}-\frac{1}{2}b^{2}\rho^{2}\alpha^{2}+b\frac{\rho\kappa}{\sigma}\alpha$
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left((1-b)^{2}\alpha^{2}-2(1-b)\alpha\eta+\eta^{2}\right)-\frac{1}{2}b^{2}\rho^{2}\alpha^{2}+b\frac{\rho\kappa}{\sigma}\alpha$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{b}{1-b}\left((1-b)^{2}\alpha^{2}-2(1-b)\alpha\eta\right)-\frac{1}{2}b^{2}\rho^{2}\alpha^{2}+b\frac{\rho\kappa}{\sigma}\alpha$
$\displaystyle=$
$\displaystyle-\frac{1}{2}b(1-b)\alpha^{2}+b\alpha\eta-\frac{1}{2}b^{2}\rho^{2}\alpha^{2}+b\frac{\rho\kappa}{\sigma}\alpha$
$\displaystyle=$ $\displaystyle
b\alpha\left[\eta-\frac{1}{2}\alpha+\frac{\rho\kappa}{\sigma}+\frac{1}{2}\alpha
b(1-\rho^{2})\right].$

  * •

Case $\pi^{\ast}(t)=\beta$:
Following the same arguments as in the case $\pi^{\ast}(t)=\alpha$ yields

$\displaystyle\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-t)\right)+\sigma\rho
B(T-t)\right)^{2}-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(t)\right)^{2}$
$\displaystyle\quad+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(t)+\frac{b}{1-b}\frac{\rho}{\sigma}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-t)\right)+\sigma\rho\right]B^{\prime}(T-t)$
$\displaystyle=$ $\displaystyle
b\beta\left[\eta-\frac{1}{2}\beta+\frac{\rho\kappa}{\sigma}+\frac{1}{2}\beta
b(1-\rho^{2})\right].$

Combining all three cases with Assumption 2.4 finally yields the claim. ∎

###### Proof of Theorem 2.9.

Let B be as in Theorem 2.5,

$A(\tau)=rb\tau+\kappa\theta\int_{0}^{\tau}B(s)ds.$

and define $G:[0,T]\times(0,\infty)^{2}\rightarrow\mathbbm{R}$ as
$G(t,v,z)=\frac{1}{b}v^{b}\exp(A(T-t)+B(T-t)z)$. Then, $A$ and $B$ are
solutions to the ODEs (7) and (8) and thus continuous and bounded on $[0,T].$
Moreover, according to Lemma 2.1, G is a solution to the HJB PDE (5). Hence,
as per Lemma 2.2, it only remains to verify that the sequence

$\big{(}G(\tau^{0}_{n,t},V^{v_{0},\pi^{\ast}}(\tau^{0}_{n,t}),z(\tau^{0}_{n,t}))\big{)}_{n\in\mathbbm{N}}$

is uniformly integrable for all $t\in[0,T]$. Let $\mathcal{T}$ denote the set
of all $\mathbb{F}$ stopping times taking values in $[0,T]$. We verify the
stronger statement that

$\big{(}G(\tau,V^{v_{0},\pi^{\ast}}(\tau),z(\tau))\big{)}_{\tau\in\mathcal{T}}$

is bounded in $L^{q}$ for some $q>1$ (see e.g. Theorem 4.6.2 in Durrett
(2019)). The argument is an adaptation of the proof of Theorem 5.3 in Kraft
(2005). Consider arbitrary $t\in[0,T]$ and $q=1+\epsilon$ with $\epsilon>0$
and define

$D(t)=bq\sqrt{1-\rho^{2}}\int_{0}^{t}\pi^{\ast}(s)\sqrt{z(s)}d\hat{W}(s).$

Then, by defining the deterministic, continuous function

$det_{1}(t)=q\left(-\ln(|b|)+b\ln(v_{0})+brt+A(T-t)\right),$

we obtain

$\displaystyle q\ln$
$\displaystyle\left(\left|G\left(t,V^{v_{0},\pi^{\ast}}(t),z(t)\right)\right|\right)$
$\displaystyle\hskip
7.11317pt=q\left(-\ln(|b|)+b\ln\left(V^{v_{0},\pi^{\ast}}(t)\right)+A(T-t)+B(T-t)z(t)\right)$
$\displaystyle\hskip
7.11317pt=q\Bigg{(}-\ln(|b|)+b\ln(v_{0})+b\int_{0}^{t}\big{(}r+\eta\pi^{\ast}(s)z(s)-\frac{1}{2}\left(\pi^{\ast}(s)\right)^{2}z(s)\big{)}ds$
$\displaystyle\hskip
7.11317pt\qquad+b\int_{0}^{t}\pi^{\ast}(s)\sqrt{z(s)}\underbrace{dW(s)}_{\mathclap{=\rho
dW^{z}(s)+\sqrt{1-\rho^{2}}d\hat{W}(s)}}+A(T-t)+B(T-t)z(t)\Bigg{)}$
$\displaystyle\hskip 7.11317pt=det_{1}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+\frac{1}{2}q^{2}b^{2}(1-\rho^{2})\int_{0}^{t}\left(\pi^{\ast}(s)\right)^{2}z(s)ds+qB(T-t)z(t)$
$\displaystyle\hskip
7.11317pt\quad+qb\rho\int_{0}^{t}\pi^{\ast}(s)\sqrt{z(s)}dW^{z}(s)+qb\int_{0}^{t}\left(\pi^{\ast}(s)\eta-\frac{1}{2}\left(\pi^{\ast}(s)\right)^{2}\right)z(s)ds$
$\displaystyle\hskip 7.11317pt=det_{1}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+qB(T-t)z(t)+qb\rho\int_{0}^{t}\pi^{\ast}(s)\sqrt{z(s)}dW^{z}(s)$
$\displaystyle\hskip
7.11317pt\quad+q\int_{0}^{t}z(s)\left\\{b\left(\pi^{\ast}(s)\eta-\frac{1}{2}\left(\pi^{\ast}(s)\right)^{2}\right)+\frac{1}{2}qb^{2}\left(1-\rho^{2}\right)\left(\pi^{\ast}(s)\right)^{2}\right\\}ds$
$\displaystyle\overset{q=1+\epsilon}{=}det_{1}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+qB(T-t)z(t)+qb\rho\int_{0}^{t}\pi^{\ast}(s)\sqrt{z(s)}dW^{z}(s)$
$\displaystyle\hskip
7.11317pt\quad+q\int_{0}^{t}z(s)\Bigg{\\{}\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}b(1-b)\left(\pi^{\ast}(s)-\frac{\eta}{1-b}\right)^{2}$
(24) $\displaystyle\hskip
78.24507pt-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(s)\right)^{2}+\frac{1}{2}\epsilon
b^{2}(1-\rho^{2})\left(\pi^{\ast}(s)\right)^{2}\Bigg{\\}}ds$
$\displaystyle\hskip 7.11317pt\overset{(\ref{eq: optimal portfolio Heston
implicit})}{=}det_{1}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+qB(T-t)z(t)+qb\rho\int_{0}^{t}\pi^{\ast}(s)\sqrt{z(s)}dW^{z}(s)$
$\displaystyle\hskip
7.11317pt\quad+q\int_{0}^{t}z(s)\Bigg{\\{}\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-s)\right)+\sigma\rho
B(T-s)\right)^{2}$ $\displaystyle\hskip
78.24507pt-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(s)\right)^{2}+\frac{1}{2}\epsilon
b^{2}(1-\rho^{2})\left(\pi^{\ast}(s)\right)^{2}\Bigg{\\}}ds$ (25)

$B$ is continuously differentiable and monotonous, since it is the solution to
an autonomous ODE (see Remark 1). Hence, $\pi^{\ast}$ is monotonous and
differentiable and has finite variation. Therefore, Itô’s product rule yields

$\displaystyle d\left(\pi^{\ast}(t)z(t)\right)$
$\displaystyle=\pi^{\ast}(t)dz(t)+z(t)d\pi^{\ast}(t)$
$\displaystyle=\pi^{\ast}(t)dz(t)-\frac{z(t)}{1-b}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-t)\right)+\sigma\rho\right]B^{\prime}(T-t)dt$
$\displaystyle=\pi^{\ast}(t)\kappa\left(\theta-z(t)\right)dt+\pi^{\ast}(t)\sigma\sqrt{z(t)}dW^{z}(t)$
$\displaystyle\quad-\frac{z(t)}{1-b}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-t)\right)+\sigma\rho\right]B^{\prime}(T-t)dt.$

Hence,

$\displaystyle\int_{0}^{t}\pi^{\ast}(s)\sqrt{z(s)}dW^{z}(s)$
$\displaystyle=\frac{1}{\sigma}\left(\pi^{\ast}(t)z(t)-\pi^{\ast}(0)z(0)\right)-\frac{\kappa}{\sigma}\int_{0}^{t}\pi^{\ast}(s)\left(\theta-z(s)\right)ds$
$\displaystyle\quad+\frac{1}{\sigma}\frac{1}{1-b}\int_{0}^{t}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-s)\right)+\sigma\rho\right]B^{\prime}(T-s)z(s)ds$
(26)

Substituting (26) in (25), while defining the deterministic and continuous
function

$det_{2}(t)=det_{1}(t)-\frac{qb\rho}{\sigma}\left[\pi^{\ast}(0)z(0)+\kappa\int_{0}^{t}\pi^{\ast}(s)\theta
ds\right],$

yields

$\displaystyle q\ln$
$\displaystyle\left(\left|G\left(t,V^{v_{0},\pi^{\ast}}(t),z(t)\right)\right|\right)$
$\displaystyle=det_{2}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+qz(t)\left[\frac{b\rho}{\sigma}\pi^{\ast}(t)+B(T-t)\right]$
$\displaystyle\quad+q\int_{0}^{t}z(s)\Bigg{\\{}\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-s)\right)+\sigma\rho
B(T-s)\right)^{2}$ $\displaystyle\hskip
71.13188pt-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(s)\right)^{2}+\frac{1}{2}\epsilon
b^{2}(1-\rho^{2})\left(\pi^{\ast}(s)\right)^{2}$ $\displaystyle\hskip
71.13188pt+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(s)+\frac{b}{1-b}\frac{\rho}{\sigma}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-s)\right)+\sigma\rho\right]B^{\prime}(T-s)\Bigg{\\}}ds$
(27)

Combining Lemma 2.7 with (27) yields

$\displaystyle q\ln$
$\displaystyle\left(\left|G\left(t,V^{v_{0},\pi^{\ast}}(t),z(t)\right)\right|\right)$
$\displaystyle\leq det_{2}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+q\frac{\kappa}{\sigma^{2}}z(t)$
$\displaystyle\quad+q\int_{0}^{t}z(s)\Bigg{\\{}\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-s)\right)+\sigma\rho
B(T-s)\right)^{2}$ $\displaystyle\hskip
71.13188pt-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(s)\right)^{2}+\frac{1}{2}\epsilon
b^{2}(1-\rho^{2})\left(\pi^{\ast}(s)\right)^{2}$ $\displaystyle\hskip
71.13188pt+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(s)+\frac{b}{1-b}\frac{\rho}{\sigma}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-s)\right)+\sigma\rho\right]B^{\prime}(T-s)\Bigg{\\}}ds$
$\displaystyle=det_{2}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+q\frac{\kappa}{\sigma^{2}}\left(z(0)+\int_{0}^{t}\kappa(\theta-z(s))ds+\sigma\int_{0}^{t}\sqrt{z}(s)dW^{z}(s)\right)$
$\displaystyle\quad+q\int_{0}^{t}z(s)\Bigg{\\{}\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-s)\right)+\sigma\rho
B(T-s)\right)^{2}$ $\displaystyle\hskip
71.13188pt-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(s)\right)^{2}+\frac{1}{2}\epsilon
b^{2}(1-\rho^{2})\left(\pi^{\ast}(s)\right)^{2}$ $\displaystyle\hskip
71.13188pt+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(s)+\frac{b}{1-b}\frac{\rho}{\sigma}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-s)\right)+\sigma\rho\right]B^{\prime}(T-s)\Bigg{\\}}ds$
$\displaystyle=\underbrace{det_{2}(t)+q\frac{\kappa}{\sigma^{2}}\left(z(0)+\int_{0}^{t}\kappa\theta
ds\right)}_{=:det_{3}(t)}+D(t)-\frac{1}{2}\langle
D\rangle_{t}+\underbrace{\frac{q\kappa}{\sigma}\int_{0}^{t}\sqrt{z(s)}dW^{z}(s)}_{=:M(t)}$
$\displaystyle\quad+q\int_{0}^{t}z(s)\Bigg{\\{}-\frac{\kappa^{2}}{\sigma^{2}}+\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-s)\right)+\sigma\rho
B(T-s)\right)^{2}$ $\displaystyle\hskip
71.13188pt-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(s)\right)^{2}+\frac{1}{2}\epsilon
b^{2}(1-\rho^{2})\left(\pi^{\ast}(s)\right)^{2}$ $\displaystyle\hskip
71.13188pt+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(s)+\frac{b}{1-b}\frac{\rho}{\sigma}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-s)\right)+\sigma\rho\right]B^{\prime}(T-s)\Bigg{\\}}ds$
$\displaystyle=det_{3}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+M(t)-\frac{1}{2}\langle M\rangle_{t}$
$\displaystyle\quad+q\int_{0}^{t}z(s)\Bigg{\\{}-\frac{\kappa^{2}}{\sigma^{2}}+\frac{1}{2}\underbrace{q}_{=1+\epsilon}\frac{\kappa^{2}}{\sigma^{2}}+\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-s)\right)+\sigma\rho
B(T-s)\right)^{2}$ $\displaystyle\hskip
71.13188pt-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(s)\right)^{2}+\frac{1}{2}\epsilon
b^{2}(1-\rho^{2})\left(\pi^{\ast}(s)\right)^{2}$ $\displaystyle\hskip
71.13188pt+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(s)+\frac{b}{1-b}\frac{\rho}{\sigma}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-s)\right)+\sigma\rho\right]B^{\prime}(T-s)\Bigg{\\}}ds$
$\displaystyle=det_{3}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+M(t)-\frac{1}{2}\langle M\rangle_{t}$ (28)
$\displaystyle\quad+q\int_{0}^{t}z(s)\Bigg{\\{}-\frac{\kappa^{2}}{2\sigma^{2}}+\frac{1}{2}\frac{b}{1-b}\eta^{2}-\frac{1}{2}\frac{b}{1-b}\left(\lambda^{\ast}\left(B(T-s)\right)+\sigma\rho
B(T-s)\right)^{2}$ $\displaystyle\hskip
71.13188pt-\frac{1}{2}b^{2}\rho^{2}\left(\pi^{\ast}(s)\right)^{2}+\frac{1}{2}\epsilon
b^{2}(1-\rho^{2})\left(\pi^{\ast}(s)\right)^{2}+\frac{1}{2}\epsilon\frac{\kappa^{2}}{\sigma^{2}}$
$\displaystyle\hskip
71.13188pt+b\frac{\rho\kappa}{\sigma}\pi^{\ast}(s)+\frac{b}{1-b}\frac{\rho}{\sigma}\left[\left(\lambda^{\ast}\right)^{\prime}\left(B(T-s)\right)+\sigma\rho\right]B^{\prime}(T-s)\Bigg{\\}}ds$
$\displaystyle=det_{3}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+M(t)-\frac{1}{2}\langle
M\rangle_{t}+q\int_{0}^{t}z(s)\Big{\\{}\left(\ast\right)\Big{\\}}ds$ (29)

According to Lemma 2.8, the expression $(\ast)$ in (29) is negative for all
$s\in[0,T]$ if $\epsilon=0$. Hence, due to the continuity of $(\ast)$ in
$\epsilon$ and the boundedness of all deterministic functions in $(\ast)$
w.r.t. $t,$ there exists $\epsilon>0$ such that $(\ast)<0$ for all
$s\in[0,T]$. For such a choice of $\epsilon$ we obtain

$\displaystyle
q\ln\left(\left|G\left(t,V^{v_{0},\pi^{\ast}}(t),z(t)\right)\right|\right)$
$\displaystyle\leq det_{3}(t)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+M(t)-\frac{1}{2}\langle M\rangle_{t}$
$\displaystyle\leq\sup_{s\in[0,T]}\left(det_{3}(s)\right)+D(t)-\frac{1}{2}\langle
D\rangle_{t}+M(t)-\frac{1}{2}\langle M\rangle_{t}$
$\displaystyle\Leftrightarrow\
\left|G\left(t,V^{v_{0},\pi^{\ast}}(t),z(t)\right)\right|^{q}$
$\displaystyle\leq\exp\left(\sup_{s\in[0,T]}\left(det_{3}(s)\right)\right)\cdot\underbrace{\exp\left(D(t)-\frac{1}{2}\langle
D\rangle_{t}\right)\cdot\exp\left(M(t)-\frac{1}{2}\langle
M\rangle_{t}\right)}_{=:(\ast\ast)}.$

Since $D$ and $M$ are local martingales with independent diffusions,
$(\ast\ast)$ is a supermartingale. Hence, Doob’s optional sampling theorem
finally yields

$\displaystyle\sup_{\tau\in\mathcal{T}}\mathbbm{E}\left[\left|G(\tau,V^{v_{0},\pi^{\ast}}(\tau),z(\tau))\right|^{q}\right]$
$\displaystyle\qquad\leq\exp\left(\sup_{s\in[0,T]}\left(det_{3}(s)\right)\right)\sup_{\tau\in\mathcal{T}}\mathbbm{E}\left[\exp\left(D(t)-\frac{1}{2}\langle
D\rangle_{t}\right)\cdot\exp\left(M(t)-\frac{1}{2}\langle
M\rangle_{t}\right)\right]$
$\displaystyle\qquad\leq\exp\left(\sup_{s\in[0,T]}\left(det_{3}(s)\right)\right)\mathbbm{E}\left[\exp\left(D(0)-\frac{1}{2}\langle
D\rangle_{0}\right)\cdot\exp\left(M(0)-\frac{1}{2}\langle
M\rangle_{0}\right)\right]$
$\displaystyle\qquad=\exp\left(\sup_{s\in[0,T]}\left(det_{3}(s)\right)\right)<\infty.$

Thus,
$\left(G(\tau,V^{v_{0},\pi^{\ast}}(\tau),z(\tau))\right)_{\tau\in\mathcal{T}}$
is bounded in $L^{q}$ for some $q>1$ and therefore uniformly integrable.
Hence, $\pi^{\ast}$ is optimal for $\mathbf{(P)}$ as per Lemma 2.2. ∎

###### Proof of Corollary 2.10.

Noting that $\delta_{K}(\lambda)=\infty$ for all $x\neq 0$ if $K=\mathbbm{R}$,
the minimum in (9) is attained by $\lambda^{\ast}(B)=0$ for all
$B\in\mathbbm{R}$ and the ODE (8) simplifies to

$\displaystyle B_{u}^{\prime}(\tau)$
$\displaystyle=-r_{0}+r_{1}B_{u}(\tau)+\frac{1}{2}r_{2}\big{(}B_{u}(\tau)\big{)}^{2}$

with coefficients

$\displaystyle r_{0}=-\frac{b}{2(1-b)}\eta^{2},\quad
r_{1}=\frac{b}{1-b}\eta\sigma\rho-\kappa,\quad
r_{2}=\sigma^{2}\left(1+\frac{b}{1-b}\rho^{2}\right).$

The expression for $\pi_{u}$ can be directly read off from Lemma 2.2. ∎

###### Proof of Lemma 2.11.

Let $\rho=0.$ Then, according to Remark 2 and Corollary 2.10, both
$\hat{\pi}^{\ast}(t)=\pi_{M}$ and $\pi_{u}(t)=\pi_{M}$ are constant in time
$t\in[0,T].$ Thus,

$\displaystyle\pi^{\ast}(t)=\text{Cap}(\hat{\pi}(t),\alpha,\beta)=\text{Cap}(\pi_{M},\alpha,\beta)=\text{Cap}(\pi_{u}(t),\alpha,\beta)\quad\forall
t\in[0,T].$

Consider now the case $\rho\neq 0$ and $\pi_{M}\in K=[\alpha,\beta].$ Then,
$\alpha(1-b)\leq\eta\leq\beta(1-b)$ and thus $B_{-}\leq 0\leq B_{+}.$
Following Lemma 2.3 and Remark 2, $0\in Z_{0}$ and the Riccati ODE (8) has
coefficients $r_{0},$ $r_{1},$ and $r_{2}$ at $\tau=0.$ Hence,
$B(\tau)=B_{u}(\tau)$ for all $\tau\in[0,T]$ such that $\rho B(\tau)\in
Z_{0}.$ Moreover, following Remark 1, $B_{u}(\tau)$ and $B(\tau)$
(respectively $\pi_{u}(t)$ and $\hat{\pi}^{\ast}(t)$) are solutions to
autonomous ODEs and thus monotone functions in $\tau$ (respectively $t$).
Noting that $\pi_{u}(t)\in K=[\alpha,\beta]\Leftrightarrow\rho B_{u}(T-t)\in
Z_{0},$ we know if $\pi_{u}(t)\in K,$ then

$\pi_{u}(t)=\frac{1}{1-b}\left(\eta+\sigma\rho
B_{u}(T-t)\right)=\frac{1}{1-b}\left(\eta+\sigma\rho
B(T-t)\right)=\hat{\pi}(t).$

Due to the monotonicity of $\pi_{u}$ and $\hat{\pi}^{\ast},$ we have
$\pi_{u}(t)\notin(\alpha,\beta)$ if and only if
$\hat{\pi}(t)\notin(\alpha,\beta).$ Thus, we have in total

$\pi^{\ast}(t)=\text{Cap}\left(\hat{\pi}^{\ast}(t),\alpha,\beta\right)=\text{Cap}\left(\pi_{u}(t),\alpha,\beta\right)\quad\forall
t\in[0,T].$

In particular, in both cases $\rho=0$ and $\pi_{M}\in K$ we have
$\mathcal{P}^{H}_{K}=\mathcal{P}^{BS}_{K}.$ ∎

###### Proof of Lemma 2.12.

The implication (ii) $\Rightarrow$ (i) is trivial, because if (ii) holds, then
$\hat{\pi}^{\ast}(t)\neq\pi_{u}(t)$ and either
$\hat{\pi}^{\ast}(t)\in(\alpha,\beta)$ or $\pi_{u}(t)\in(\alpha,\beta).$
Hence,

$\pi^{\ast}(t)=\text{Cap}\left(\hat{\pi}^{\ast}(t),\alpha,\beta\right)\neq\text{Cap}\left(\pi_{u}(t),\alpha,\beta\right),$

and therefore
$\mathcal{P}^{H}_{K}\left(\pi_{u},\alpha,\beta\right)\neq\mathcal{P}^{BS}_{K}\left(\pi_{u},\alpha,\beta\right).$
Assume now that (i) holds. According to Lemma 2.11, this implies $\rho\neq 0$
and $\pi_{M}\notin K=[\alpha,\beta].$ Moreover, as $B$ and $B_{u}$ are
continuously differentiable,

$\hat{\pi}^{\ast}(t)=\frac{1}{1-b}\left(\eta+\sigma\rho
B(T-t)\right)\quad\text{and}\quad\pi_{u}(t)=\frac{1}{1-b}\left(\eta+\sigma\rho
B_{u}(T-t)\right)$

are continuously differentiable functions with
$\hat{\pi}^{\ast}(T)=\pi_{u}(T)=\pi_{M}\notin K.$ Thus, (i) can only be
satisfied if there exists $0<t<T$ such that

$\displaystyle\left\\{\hat{\pi}^{\ast}(t),\pi_{u}(t)\right\\}\cap
K\neq\emptyset,$ (30)

as otherwise
$\pi^{\ast}(t)=\text{Cap}(\hat{\pi}^{\ast}(t),\alpha,\beta)=\text{Cap}(\pi_{M},\alpha,\beta)=\text{Cap}(\pi_{u}(t),\alpha,\beta)$
for all $t\in[0,T].$ Thus, we define the latest such time point as

$\displaystyle\hat{t}=\sup\left\\{0\leq t\leq T\ \Big{|}\
\left\\{\hat{\pi}^{\ast}(t),\pi_{u}(t)\right\\}\cap K\neq\emptyset\right\\}.$

Since there exists a positive $t>0$ satisfying (30), we know that $\hat{t}>0.$
Moreover, as $\pi_{M}\notin K$ and $K$ is closed, we also know that
$\hat{t}<T.$ If $\hat{\pi}^{\ast}(\hat{t})\neq\pi_{u}(\hat{t}),$ then only one
of these portfolio processes takes a value in $K$ and (ii) holds for
$\hat{t}$.
It thus remains to verify that $\hat{\pi}^{\ast}(\hat{t})\neq\pi_{u}(\hat{t})$
to conclude the proof. This will be proven by contradiction. Assume that
$\hat{\pi}^{\ast}(\hat{t})=\pi_{u}(\hat{t})\in K$ which is equivalent to $\rho
B(T-\hat{t})=\rho B_{u}(T-\hat{t})\in Z_{0}.$ Further,

$\displaystyle T-\hat{t}=\inf\left\\{0\leq\tau\leq T\ \Big{|}\ \left\\{\rho
B(\tau),\rho B_{u}(\tau)\right\\}\cap Z_{0}\neq\emptyset\right\\}.$

Moreover, as $\pi_{M}\notin K\overset{\rho\neq 0}{\Leftrightarrow}\rho
B(0)=\rho B_{u}(0)=0\notin Z_{0}$, $B$ and $B_{u}$ are non-constant and thus
strictly monotone functions. However, this implies that at $\tau=T-\hat{t},$
$\rho B(\tau)$ and $\rho B_{u}(\tau)$ have the same value at the boundary of
$Z_{0}$ and satisfy the same ODE while taking values in $Z_{0}.$ Hence,
$B(\tau)=B_{u}(\tau)$ for all $\tau\in[0,T]$ such that $B(\tau)\in Z_{0}.$
Noting that

$\rho B(\tau)\in Z_{0}\Leftrightarrow\hat{\pi}^{\ast}(\tau)\in
K\quad\text{and}\quad\rho B_{u}(\tau)\in Z_{0}\Leftrightarrow\pi_{u}(\tau)\in
K,$

finally yields for all $t\in[0,T]$:

$\displaystyle\pi^{\ast}(t)$
$\displaystyle=\text{Cap}\left(\hat{\pi}^{\ast}(t),\alpha,\beta\right)=\text{Cap}\Big{(}\frac{1}{1-b}(\eta+\sigma\underbrace{\rho
B(T-t)}_{\mathclap{=\rho B_{u}(T-t)\ \text{while }\hat{\pi}^{\ast}(t)\in
K}}),\alpha,\beta\Big{)}$
$\displaystyle=\text{Cap}\left(\frac{1}{1-b}(\eta+\sigma\rho
B_{u}(T-t)),\alpha,\beta\right)=\text{Cap}\left(\pi_{u}(t),\alpha,\beta\right),$

which is a contradiction to (i). Thus,
$\hat{\pi}^{\ast}(\hat{t})\neq\pi_{u}(\hat{t}).$ ∎

###### Proof of Corollary 2.13.

123
Proof of (i): Let $\pi_{M}<\alpha$ and
$0<\alpha=2\pi_{M}<\alpha$$\frac{2}{1-b}\eta.$ Then, $\eta>0,$ $\pi_{M}>0$ and
$\eta=(1-b)\pi_{M}<(1-b)\alpha\Leftrightarrow B_{-}>0.$ Following Lemma 2.3
and Remark 2, $0\in Z_{-}$ and the Riccati ODE (8) has coefficients
$r_{0}^{-},$ $r_{1}^{-},$ and $r_{2}^{-}$ at $\tau=0.$ However, due to the
initial condition $B(0)=0$, this implies

$\displaystyle
B^{\prime}(0)=-r^{-}_{0}+r^{-}_{1}B(0)+\frac{1}{2}r^{-}_{2}B(0)^{2}=-r^{-}_{0}=-\frac{1}{2}b\alpha\left((1-b)\alpha-2\eta\right)=0,$

i.e., according to Remark 1, $B$ is constant with $B(\tau)=0$ for all
$\tau\in[0,T].$ In particular, for all $t\in[0,T]$ we have
$\hat{\pi}^{\ast}(t)=\pi_{M}$ and

$\pi^{\ast}(t)=\text{Cap}(\hat{\pi}^{\ast}(t),\alpha,\beta)=\text{Cap}(\pi_{M},\alpha,\beta)=\alpha.$

On the other hand, the optimal unconstrained portfolio $\pi_{u}$ does not
violate the constraint at time $t^{\ast}$, i.e.

$\text{Cap}(\pi_{u}(t^{\ast}),\alpha,\beta)=\pi_{u}(t^{\ast})\in(\alpha,\beta),$

and thus $\mathcal{P}^{H}_{K}\neq\mathcal{P}^{BS}_{K}.$
Proof of (ii): According to Corollary 2.10,

$\frac{\partial}{\partial
t}\left(\pi_{u}(t)\right)=\frac{1}{1-b}\frac{\partial}{\partial
t}\left(\eta+\sigma\rho B_{u}(T-t)\right)$

Since $B_{u}$ is the solution to the autonomous ODE (16), $B_{u}$ is a
monotone function (cf. Remark 1). Therefore,

$\displaystyle\text{sign}\left(\frac{\partial}{\partial
t}\left(\pi_{u}(t)\right)\right)$ $\displaystyle\hskip 2.84526pt=\hskip
2.84526pt\text{sign}\left(\frac{-\sigma\rho}{1-b}B_{u}^{\prime}(T-t)\right)=\text{sign}\left(\frac{-\sigma\rho}{1-b}B_{u}^{\prime}(0)\right)\overset{(\ref{eq:
Unconstr ODE B})}{=}\text{sign}\left(\frac{\sigma\rho}{1-b}r_{0}\right)$
$\displaystyle\overset{(\ref{eq: ODE
zones})}{=}\text{sign}\left(\frac{\sigma\rho}{1-b}\left(-\frac{b}{2(1-b)}\eta^{2}\right)\right)$

As $\frac{\sigma\eta^{2}}{2(1-b)^{2}}$ is positive, this implies
$\text{sign}\left(\frac{\partial}{\partial
t}\left(\pi_{u}(t)\right)\right)=-\text{sign}\left(b\rho\right).$ As $\beta>0$
and $\pi_{M}>\beta\Leftrightarrow 0=\rho B(0)>B_{+},$ we follow the same line
of argument for $\hat{\pi}^{\ast}$ to obtain

$\displaystyle\text{sign}\left(\frac{\partial}{\partial
t}\left(\hat{\pi}^{\ast}(t)\right)\right)$
$\displaystyle=\text{sign}\left(\frac{-\sigma\rho}{1-b}B^{\prime}(0)\right)=\text{sign}\left(\frac{\sigma\rho}{1-b}r_{0}^{+}\right)$
$\displaystyle=\text{sign}\left(\frac{\sigma\rho}{1-b}\frac{1}{2}b\beta\left([1-b]\beta-2\eta\right)\right)=\text{sign}\left(\frac{\sigma\rho
b\beta}{2}\left(\beta-2\pi_{M}\right)\right)$

Disregarding the positive factor $\frac{\sigma\beta}{2}$ finally yields

$\displaystyle\text{sign}\left(\frac{\partial}{\partial
t}\left(\hat{\pi}^{\ast}(t)\right)\right)=\text{sign}\big{(}\rho
b\underbrace{\left(\beta-2\pi_{M}\right)}_{<0}\big{)}=-\text{sign}\left(b\rho\right)=\text{sign}\left(\frac{\partial}{\partial
t}\left(\pi_{u}(t)\right)\right).$

If now $\rho<0$ and $b<0,$ then both portfolio allocation
$\pi^{\ast}(t)=\text{Cap}\left(\hat{\pi}^{\ast}(t),\alpha,\beta\right)$ and
$\text{Cap}\left(\pi_{u}(t),\alpha,\beta\right)$ are non-increasing in time.
In particular, as both allocations are equal to $\beta<\pi_{M}$ at $t=T,$ they
must be equal (to $\beta$) throughout the entire investment horizon. Hence,
the projections $\mathcal{P}^{H}_{K}$ and $\mathcal{P}^{BS}_{K}$ coincide
under these assumptions. ∎

###### Proof of Theorem 3.1.

We transform the portfolio optimisation problem $\mathbf{(P)}$ into an
equivalent optimisation problem by applying the change of control
$\pi_{A}(t):=A^{\prime}\pi(t)$ for any given $\pi\in\Lambda^{PCSV}.$
Expressing the SDE of the wealth process $V^{v_{0},\pi}$ in terms of $\pi_{A}$
yields

$\displaystyle dV^{v_{0},\pi}(t)$
$\displaystyle=V^{v_{0},\pi}(t)V^{v_{0},\pi}(t)\left[\left(r+\underbrace{\eta^{\prime}A}_{=:\eta_{A}^{\prime}}\text{diag}(z(t))\underbrace{A^{\prime}\pi(t)}_{=\pi_{A}(t)}\right)dt+\underbrace{\pi(t)^{\prime}A}_{=\pi_{A}(t)^{\prime}}\text{diag}(\sqrt{z(t)})dW(t)\right]$
$\displaystyle=V^{v_{0},\pi}(t)\left[\left(r+\eta_{A}^{\prime}\text{diag}(z(t))\pi_{A}(t)\right)dt+\pi_{A}(t)^{\prime}\text{diag}(\sqrt{z(t)})dW(t)\right]$
(31)

and $\pi(t)\in
K_{PCSV}\Leftrightarrow\pi_{A}(t)\in\bigtimes_{i=1}^{d}[0,\sqrt{\beta_{i}}].$
In particular, we may equivalently rewrite the portfolio optimisation problem
in terms of $\pi_{A}$ as

$\displaystyle\mathbf{(P_{A})}\begin{cases}\Phi(v_{0})=\underset{\pi\in\Lambda_{A}}{\sup}\mathbbm{E}\big{[}U(V^{v_{0},\pi}(T))\big{]}\\\
\Lambda_{A}=\big{\\{}\pi_{A}\in\Lambda_{PCSV}\ \big{|}\
\pi_{A}(t)\in\bigtimes_{i=1}^{d}[0,\sqrt{\beta_{i}}]\ \mathcal{L}[0,T]\otimes
Q-\text{a.e.}\\}\end{cases}.$

We proceed by solving $\mathbf{(P_{A})}$ and then inverting the change of
control to obtain a solution for the original optimisation problem
$\mathbf{(P)}.$ As $K_{A}=\bigtimes_{i=1}^{d}[0,\sqrt{\beta_{i}}]$ is a
d-dimensional interval, $\mathbf{(P_{A})}$ fits into the setting of Section
4.2 in Escobar-Anel et al. (2023) with $m=d$ and $\Sigma_{i}=1$ for
$i=1,..,d.$ According to Lemma 4.5 in Escobar-Anel et al. (2023), if
$A:[0,T]\rightarrow\mathbbm{R},$ $B:[0,T]\rightarrow\mathbbm{R}^{d}$ with
$A(0)=B_{i}(0)=0$ for $i=1,...,d$ satisfy

$\displaystyle A^{\prime}(\tau)$
$\displaystyle=br+\sum_{i=1}^{d}\kappa_{i}\theta_{i}$ $\displaystyle
B_{i}^{\prime}(\tau)$
$\displaystyle=-\kappa_{i}B_{i}(\tau)+\frac{1}{2}\sigma_{i}\left(B_{i}(\tau)\right)^{2}+\frac{1}{2}\frac{b}{1-b}\inf_{\lambda\in\mathbbm{R}}\left\\{2(1-b)\delta_{[0,\sqrt{\beta_{i}}]}(\lambda)+\left(\left(\eta_{A}\right)_{i}+\lambda_{i}+\sigma_{i}\rho_{i}B_{i}(\tau)\right)^{2}\right\\},$
(32)

then the function
$G(t,v,z):=\frac{1}{b}v^{b}\exp\left(A(T-t)+B(T-t)^{\prime}z\right)$ is a
solution to the HJB PDE associated with $\mathbf{(P_{A})}.$ However, for each
$i=1,...,d,$ equation (32) contains the same ODE and optimisation that were
considered in Lemma 2.3 and Section 2. Since Assumption 2.4 is satisfied, the
solution $B_{i}$ to (32) is given by Theorem 2.5. Finally, the candidate
optimal portfolio $\pi^{\ast}_{A}$ for $\mathbf{(P)}$ is given by

$\displaystyle\left(\pi^{\ast}_{A}\right)_{i}(t)=\text{Cap}\left(\frac{1}{b}\left(\left(\eta_{A}\right)_{i}+\sigma_{i}\rho_{i}B_{i}(T-t)\right),0,\sqrt{\beta_{i}}\right).$

We still need to formally verify the optimality of $\pi^{\ast}_{A}$ for
$\mathbf{(P_{A})}.$ For this purpose, we define the sequence of stopping times
$\tau_{n,t}$ as $\tau_{n,t}=\min(T,\hat{\tau}_{n,t})$, with

$\displaystyle\hat{\tau}_{n,t}=\inf\Big{\\{}t\leq u\leq T\ \Big{|}\ $
$\displaystyle\int_{t}^{u}\left\|b\left(\sqrt{z(s)}\odot\pi_{A}(s)\right)G(s,V^{v_{0},\pi^{\ast}}(s),z(s))\right\|^{2}ds\geq
n,$ $\displaystyle\int_{t}^{u}\left\|\left(\sigma\odot\sqrt{z(s)}\odot
B(T-s)\right)G(s,V^{v_{0},\pi^{\ast}}(s),z(s))\right\|^{2}ds\geq n\Big{\\}}.$

According to Theorem 3.12 in Escobar-Anel et al. (2023), the optimality of
$\pi^{\ast}_{A}$ is verified if

$\left(G\left(\tau_{n,t},V^{v_{0},\pi^{\ast}}(\tau_{n,t}),z(\tau_{n,t})\right)\right)_{n\in\mathbbm{N}}$

is uniformly integrable for every $t\in[0,T].$ However, as Assumptions 2.4 and
2.6 are satisfied for every $i=1,...,d,$ following the same steps as in the
proof of Theorem 2.9, we can show that there exists a constant $q>1$ and a
bounded, deterministic function $det(t)$ such that the local martingales

$\displaystyle D_{i}(t)$
$\displaystyle=bq\sqrt{1-\rho_{i}^{2}}\int_{0}^{t}\left(\pi_{A}\right)_{i}^{\ast}(s)\sqrt{z_{i}(s)}d\hat{W}_{i}(s)$
$\displaystyle M_{i}(t)$
$\displaystyle=\frac{q\kappa_{i}}{\sigma_{i}}\int_{0}^{t}\sqrt{z_{i}(s)}dW^{z}_{i}(s)$

can be used to bound
$\left|G\left(t,V^{v_{0},\pi^{\ast}}(t),z(t)\right)\right|^{q}$ for every
$t\in[0,T]$ through

$\displaystyle\Big{|}G$
$\displaystyle\left(t,V^{v_{0},\pi^{\ast}}(t),z(t)\right)\Big{|}^{q}\leq\exp\left(\sup_{s\in[0,T]}det(s)\right)\underbrace{\exp\left(\sum_{i=1}^{d}D_{i}(t)-\frac{1}{2}\langle
D_{i}\rangle_{t}+M_{i}(t)-\frac{1}{2}\langle
M_{i}\rangle_{t}\right)}_{(\ast)}.$

The diffusions of the local martingales $D_{i},$ $M_{i}$ are independent and
therefore $(\ast)$ is a supermartingale. Since the above bound holds pointwise
for every $t\in[0,T],$ Doob’s optional sampling theorem yields the
$L^{q}$-boundedness (and thereby uniform integrability) of

$\Big{(}G(\tau^{0}_{n,t},V^{v_{0},\pi^{\ast}}(\tau^{0}_{n,t}),z(\tau^{0}_{n,t}))\Big{)}_{n\in\mathbbm{N}}.$

Hence, $\pi_{A}^{\ast}$ is optimal for $\mathbf{(P_{A})}$ and thus
$\pi^{\ast}(t)=A\pi^{\ast}_{A}(t)$ is optimal for $\mathbf{(P)}$ in
$\mathcal{M}_{PCSV}.$ ∎

###### Proof of Theorem 3.2.

The proof follows similar arguments as the proof of Corollary 5.4 in Kraft
(2005). For given $\pi\in\Lambda^{\gamma,\Sigma},$ let $\hat{\pi}_{BS}$ and
$\hat{\pi}_{H}$ be defined through

$\displaystyle\hat{\pi}_{BS}(t):=\Sigma(z(t))\pi(t)\qquad\text{or}\qquad\hat{\pi}_{H}(t):=\frac{\Sigma(z(t))}{\sqrt{z(t)}}\pi(t).$
(33)

Proof of (i): By construction, $\pi\in\Lambda_{K(\cdot)}$ if and only if
$\hat{\pi}_{BS}(t)\in\hat{K}:=[\alpha,\beta]$ $\mathcal{L}[0,T]\otimes
Q$-a.e., i.e. the constraint on $\hat{\pi}_{BS}$ is constant. Under the
present assumptions, the wealth process satisfies

$\displaystyle dV^{v_{0},\pi}(t)$
$\displaystyle=V^{v_{0},\pi}(t)\left([r+\eta\Sigma(z(t))\pi(t)]dt+\pi(t)\Sigma(z(t))dW(t)\right)$
$\displaystyle=V^{v_{0},\pi}(t)\left([r+\eta\hat{\pi}_{BS}(t)]dt+\hat{\pi}_{BS}(t)dW(t)\right).$

Hence, maximising $\mathbbm{E}\left[U\left(V^{v_{0},\pi}(T)\right)\right]$
over $\hat{\pi}_{BS}$ subject to the constraint
$\hat{\pi}_{BS}(t)\in\hat{K}=[\alpha,\beta]$ $\mathcal{L}[0,T]\otimes
Q\text{-a.e.}$ is equivalent to the constrained portfolio optimisation problem
in a Black-Scholes market $\mathcal{M}_{BS}$ with market price of risk $\eta$
and volatility $1.$ Therefore, as discussed in Section 2.3, the constant-mix
strategy

$\displaystyle\hat{\pi}^{\ast}_{BS}=\text{Cap}\left(\pi_{M},\alpha,\beta\right)\Leftrightarrow\pi^{\ast}(t)=\Sigma(z(t))^{-1}\text{Cap}\left(\pi_{M},\alpha,\beta\right)$

maximises the expected utility over all admissible $\hat{\pi}_{BS}.$ Inverting
the change of control through multiplication by $\Sigma(z(t))^{-1}$ yields the
claim.
Proof of (ii): By construction, $\pi\in\Lambda_{K(\cdot)}$ if and only if
$\hat{\pi}_{H}(t)\in\hat{K}:=[\alpha,\beta]$ $\mathcal{L}[0,T]\otimes Q$-a.e.,
i.e. the constraint on $\hat{\pi}_{H}$ is constant. Under the present
assumptions, the wealth process satisfies

$\displaystyle dV^{v_{0},\pi}(t)$
$\displaystyle=V^{v_{0},\pi}(t)\left([r+\eta\sqrt{z(t)}\Sigma(z(t))\pi(t)]dt+\pi(t)\Sigma(z(t))dW(t)\right)$
$\displaystyle=V^{v_{0},\pi}(t)\left([r+\eta
z(t)\hat{\pi}_{H}(t)]dt+\hat{\pi}_{H}(t)\sqrt{z(t)}dW(t)\right).$

Hence, maximising $\mathbbm{E}\left[U\left(V^{v_{0},\pi}(T)\right)\right]$
over $\hat{\pi}_{H}$ subject to the constraint
$\hat{\pi}_{H}(t)\in\hat{K}=[\alpha,\beta]$ $\mathcal{L}[0,T]\otimes
Q\text{-a.e.}$ is equivalent to the constrained portfolio optimisation problem
in the Heston market $\mathcal{M}_{H},$ as considered in Section 2. As all
requirements of Theorem 2.9 are satisfied, $\pi^{\ast}$ (as defined in Theorem
2.9) maximises the expected utility over all admissible $\hat{\pi}_{H}.$
Inverting the change of control through multiplication by
$\sqrt{z(t)}\Sigma(z(t))^{-1}$ yields the claim. ∎

##  Appendix B Supplementary Results

######  Lemma B.1.

  * (i)

Let $f:\mathbbm{R}\rightarrow\mathbbm{R}$ be a convex function. Then, for
every $x_{0}\in\mathbbm{R},$ $\epsilon>0,$ there exists an $L>0$ such that if
$|x-x_{0}|<\epsilon,$ $|y-x_{0}|<\epsilon,$ then

$|f(x)-f(y)|<L|x-y|.$

In particular, f is locally Lipschitz continuous.

  * (ii)

Consider an interval $I\subset\mathbbm{R}$ and let
$f_{1},f_{2}:I\rightarrow\mathbbm{R}$ be Lipschitz continuous on $I.$ Then,
$f:=min(f_{1},f_{2})$ is Lipschitz continuous on $I.$

###### Proof of Lemma B.1.

1
Proof of (i): Since $f$ is convex on $\mathbbm{R},$ f is continuous and
thereby bounded on any compact subset of $\mathbbm{R}.$ In particular, there
exists $m,M\in\mathbbm{R}$ such that

$m\leq f(x)\leq M\quad\text{for all }x\ \text{with }|x-x_{0}|\leq\epsilon.$

Hence, the statement follows from Section A, Lemma 3.1.1 in Hiriart-Urruty and
Lemaréchal (2001).
Proof of (ii): Let $L_{1},L_{2}$ denote the Lipschitz constant of $f_{1},$
$f_{2}$ respectively. For any real numbers $x,y\in\mathbbm{R}$ recall the
identity

$\displaystyle\min(x,y)=\frac{1}{2}\left(x+y-|x-y|\right).$ (34)

Therefore, for any $x,y\in I$ we have

$\displaystyle|f(x)-f(y)|$ $\displaystyle\hskip 2.84526pt=\hskip
2.84526pt\Big{|}\min\left(f_{1}(x),f_{2}(x)\right)-\min\left(f_{1}(y),f_{2}(y)\right)\Big{|}$
$\displaystyle\overset{(\ref{eq: identity
minimum})}{=}\frac{1}{2}\Big{|}f_{1}(x)+f_{2}(x)-|f_{1}(x)-f_{2}(x)|-f_{1}(y)-f_{2}(y)+|f_{1}(y)-f_{2}(y)|\Big{|}$
$\displaystyle\hskip 2.84526pt\leq\hskip
2.84526pt\frac{1}{2}\left(|f_{1}(x)-f_{1}(y)|+|f_{2}(x)-f_{2}(y)|+\Big{|}|f_{1}(x)-f_{2}(x)|-|f_{1}(y)-f_{2}(y)|\Big{|}\right)$
$\displaystyle\hskip 2.84526pt\leq\hskip
2.84526pt\frac{1}{2}\big{(}|f_{1}(x)-f_{1}(y)|+|f_{2}(x)-f_{2}(y)|+|f_{1}(x)-f_{2}(x)-\left(f_{1}(y)-f_{2}(y)\right)|\big{)}$
$\displaystyle\hskip 2.84526pt=\hskip
2.84526pt\frac{1}{2}\big{(}|f_{1}(x)-f_{1}(y)|+|f_{2}(x)-f_{2}(y)|+\underbrace{\left|f_{1}(x)-f_{2}(x)-f_{1}(y)+f_{2}(y)\right|}_{\leq|f_{1}(x)-f_{1}(y)|+|f_{2}(x)-f_{2}(y)|}\big{)}$
$\displaystyle\hskip 2.84526pt\leq\hskip
2.84526pt\underbrace{|f_{1}(x)-f_{1}(y)|}_{\leq
L_{1}|x-y|}+\underbrace{|f_{2}(x)-f_{2}(y)|}_{\leq L_{2}|x-y|}$
$\displaystyle\hskip 2.84526pt\leq\hskip 2.84526pt(L_{1}+L_{2})|x-y|.$

Hence, $f$ is Lipschitz continuous on $I$ with Lipschitz constant
$L=L_{1}+L_{2}.$ ∎

######  Theorem B.2.

Let $f:\mathbbm{R}\rightarrow\mathbbm{R}$ be locally Lipschitz continuous on
$\mathbbm{R}$ and $\tau_{0},B_{0}\in\mathbbm{R}$ be constants. Consider the
ordinary differential equation

$\displaystyle B^{\prime}(\tau)=f(B(\tau)),\quad B(\tau_{0})=B_{0}.$ (35)

Then there exists an $\epsilon>0$ such that the ODE (35) has a unique solution
$B(\tau)$ for $\tau\in(\tau_{0}-\epsilon,\tau_{0}+\epsilon).$

###### Proof of B.2.

This statement follows directly from Chapter §6, ‘VII. Existence and
Uniqueness Theorem’ in Walter (1998). ∎

######  Lemma B.3.

Consider the setting of Theorem B.2 and an open interval $I\subset\mathbbm{R}$
with $\tau_{0}\in I,$ and let $B:I\rightarrow\mathbbm{R}$ satisfy ODE (35).
Then, $B$ is a monotone function in $\tau.$ If $f(B_{0})\neq 0,$ then $B$ is
strictly monotone in $\tau.$ Moreover, if $f(B_{0})=0,$ then $B$ is constant
in $\tau.$

###### Proof of Lemma B.3.

We first consider the case $f(B_{0})\neq 0.$ The following argument proceeds
by contradiction. Assume that $B$ is not a strictly monotone function of
$\tau.$ Since $f$ is locally Lipschitz continuous and therefore continuous,
$B$ is continuously differentiable and there must exist $\bar{\tau}\in I$ with
$B(\bar{\tau})=\bar{B}$ such that

$B^{\prime}(\bar{\tau})=f(\bar{B})=0\text{ and }B^{\prime}(\tau)\neq 0\text{
for all }\tau\text{ with }|\tau-\tau_{0}|<|\bar{\tau}-\tau_{0}|.$

In particular, $B$ is also a solution to the ODE

$\displaystyle B^{\prime}(\tau)=f(B(\tau)),\quad B(\bar{\tau})=\bar{B}.$ (36)

On the other hand, as $f(\bar{B})=f(B(\bar{\tau}))=B^{\prime}(\bar{\tau})=0,$
the function $\hat{B}(\tau):=\bar{B}$ is also a solution to the ODE (36).
However, for all $\tau$ with $|\tau-\tau_{0}|<|\bar{\tau}-\tau_{0}|$ we know
that $f(B(\tau))=B^{\prime}(\tau)\neq 0$ and thus $B(\tau)\neq\hat{B}(\tau).$
This is a contradiction to Theorem B.2. Therefore, $B$ must be strictly
monotone in $\tau.$
If $f(B_{0})=0,$ by an analogous argument, $B(\tau)=B_{0}$ is the unique
solution to ODE (35) and therefore $B$ is constant in $\tau.$ ∎

######  Lemma B.4.

Consider a terminal time point $T>0$, real coefficients $B_{0}$, $r_{0},\
r_{1},\ r_{2}$ and the following Riccati-ODE

$\displaystyle
B^{\prime}(\tau)=-r_{0}+r_{1}B(\tau)+\frac{1}{2}r_{2}B(\tau)^{2},\quad
B(0)={B_{0}},$ (37)

where $r_{2}\neq 0,$ $r_{1}^{2}+2r_{0}r_{2}>0$ and define
$r_{3}=\sqrt{r_{1}^{2}+2r_{0}r_{2}}.$

  * (i)

The function

$\displaystyle
B(\tau)=\frac{2r_{2}r_{3}B_{0}+\left(e^{r_{3}\tau}-1\right)(r_{1}+r_{3})\left(r_{1}+r_{2}B_{0}-r_{3}\right)}{2r_{2}r_{3}-r_{2}\left(e^{r_{3}\tau}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)}$
(38)

is the unique solution of equation (37) on its maximal interval
$[0,t_{+}(B_{0}))$ with life-time $t_{+}(B_{0})>0.$ Moreover, for any
$T\in[0,t_{+}(B_{0}))$ we have

$\displaystyle\textasciiacute\int_{0}^{T}B(\tau)d\tau=\frac{2}{r_{2}}\ln\left(\frac{2r_{3}e^{\frac{r_{3}-B_{0}}{2}T}}{r_{3}\left(e^{r_{3}T}+1\right)-r_{1}\left(e^{r_{3}T}-1\right)-r_{2}\left(e^{r_{3}T}-1\right)B_{0}}\right).$
(39)

  * (ii)

The life time $t_{+}(B_{0})$ of $B$ (in the sense of Lemma 10.1 in Filipovic
(2009)) is given as

$\displaystyle t_{+}(B_{0})$
$\displaystyle=\begin{cases}\frac{1}{r_{3}}\ln\left(\frac{r_{1}+r_{2}B_{0}+r_{3}}{r_{1}+r_{2}B_{0}-r_{3}}\right),&\quad\text{if}\
r_{1}+r_{2}B_{0}-r_{3}>0\\\ \infty,&\quad\text{if}\ r_{1}+r_{2}B_{0}-r_{3}\leq
0.\\\ \end{cases}$

###### Proof of Lemma B.4.

Statement (i) follows directly from Lemma 10.12 in Filipovic (2009) with
$A=\frac{1}{2}r_{2},$ $B=r_{1},$ $C=r_{0},$ $u=B_{0}$ and

$\displaystyle-\frac{2r_{0}\left(e^{r_{3}\tau}-1\right)-\left(r_{3}\left(e^{r_{3}\tau}+1\right)+r_{1}\left(e^{r_{3}\tau}-1\right)\right)B_{0}}{r_{3}\left(e^{r_{3}\tau}+1\right)-r_{1}\left(e^{r_{3}\tau}-1\right)-r_{2}\left(e^{r_{3}\tau}-1\right)B_{0}}$
$\displaystyle\quad=-\frac{-2r_{3}B_{0}+\left(e^{r_{3}\tau}-1\right)\left(2r_{0}-\left(r_{1}+r_{3}\right)B_{0}\right)}{2r_{3}-\left(e^{r_{3}\tau}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)}$
$\displaystyle\quad=\frac{2r_{2}r_{3}B_{0}-\left(e^{r_{3}\tau}-1\right)\big{(}\overbrace{2r_{0}r_{2}}^{=r_{3}^{2}-r_{1}^{2}}-\left(r_{1}+r_{3}\right)r_{2}B_{0}\big{)}}{2r_{2}r_{3}-r_{2}\left(e^{r_{3}\tau}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)}$
$\displaystyle\quad=\frac{2r_{2}r_{3}B_{0}-\left(e^{r_{3}\tau}-1\right)\left(\left(r_{3}-r_{1}\right)\left(r_{3}+r_{1}\right)-\left(r_{1}+r_{3}\right)r_{2}B_{0}\right)}{2r_{2}r_{3}-r_{2}\left(e^{r_{3}\tau}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)}$
$\displaystyle\quad=\frac{2r_{2}r_{3}B_{0}+\left(e^{r_{3}\tau}-1\right)(r_{1}+r_{3})\left(r_{1}+r_{2}B_{0}-r_{3}\right)}{2r_{2}r_{3}-r_{2}\left(e^{r_{3}\tau}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)}.$

The function $B$ as in (38) is well-defined and continuously differentiable as
long as the denominator in (38) is non-zero. In particular, taking derivatives
verifies that $B$ satisfies (37) as long as its denominator is non-zero. This
is true for $\tau=0,$ when the denominator is equal to $r_{2}r_{3}>0,$ and the
denominator stays positive if $r_{1}+r_{2}B_{0}-r_{3}\leq 0,$ i.e.
$t_{+}(B_{0})=\infty$. However, if $r_{1}+r_{2}B_{0}-r_{3}>0,$ then

$\displaystyle\quad 0$ $\displaystyle\overset{!}{=}\quad$ $\displaystyle
2r_{2}r_{3}-r_{2}\left(e^{r_{3}t_{+}(B_{0})}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)$
$\displaystyle\Leftrightarrow$ $\displaystyle\quad 2r_{3}$
$\displaystyle=\quad$
$\displaystyle\left(e^{r_{3}t_{+}(B_{0})}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)$
$\displaystyle\Leftrightarrow$ $\displaystyle\quad t_{+}(B_{0})$
$\displaystyle=\quad$
$\displaystyle\frac{1}{r_{3}}\ln\left(\frac{r_{1}+r_{2}B_{0}+r_{3}}{r_{1}+r_{2}B_{0}-r_{3}}\right).$

∎

######  Corollary B.5.

Consider the setting of Lemma B.4, let $B$ be as in (38) and let
$\hat{B}\in\mathbbm{R}$ be given.
If

$\displaystyle\tau_{\hat{B}}:=\frac{1}{r_{3}}\ln{\left(\frac{2r_{2}r_{3}\left(\hat{B}-B_{0}\right)+\left(r_{1}+r_{2}B_{0}-r_{3}\right)\left(r_{1}+\hat{B}r_{2}+r_{3}\right)}{\left(r_{1}+r_{2}B_{0}-r_{3}\right)\left(r_{1}+\hat{B}r_{2}+r_{3}\right)}\right)}\leq
t_{+}(B_{0})$

then $B(\tau_{\hat{B}})=\hat{B}$.

###### Proof.

If $\tau_{\hat{B}}<t_{+}(B_{0})$, then

$\displaystyle
B(\tau_{\hat{B}})=\frac{2r_{2}r_{3}B_{0}+\left(e^{r_{3}\tau_{\hat{B}}}-1\right)(r_{1}+r_{3})\left(r_{1}+r_{2}B_{0}-r_{3}\right)}{2r_{2}r_{3}-r_{2}\left(e^{r_{3}\tau_{\hat{B}}}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)}.$

This in turn implies

$\displaystyle\hat{B}\overset{!}{=}B(\tau_{\hat{B}})=\frac{2r_{2}r_{3}B_{0}+\left(e^{r_{3}\tau_{\hat{B}}}-1\right)(r_{1}+r_{3})\left(r_{1}+r_{2}B_{0}-r_{3}\right)}{2r_{2}r_{3}-r_{2}\left(e^{r_{3}\tau_{\hat{B}}}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)}$
$\displaystyle\Leftrightarrow\quad$
$\displaystyle\hat{B}\left(2r_{2}r_{3}-r_{2}\left(e^{r_{3}\tau_{\hat{B}}}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)\right)$
$\displaystyle\quad=2r_{2}r_{3}B_{0}+r_{2}\left(e^{r_{3}\tau_{\hat{B}}}-1\right)\left(r_{1}+r_{3}\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)$
$\displaystyle\Leftrightarrow\quad$ $\displaystyle
2r_{2}r_{3}\left(\hat{B}-B_{0}\right)=\left(e^{r_{3}\tau_{\hat{B}}}-1\right)\left(r_{1}+r_{2}B_{0}-r_{3}\right)\left(r_{1}+\hat{B}r_{2}+r_{3}\right)$
$\displaystyle\Leftrightarrow\quad$
$\displaystyle\left(e^{r_{3}\tau_{\hat{B}}}-1\right)=\frac{2r_{2}r_{3}\left(\hat{B}-B_{0}\right)}{\left(r_{1}+r_{2}B_{0}-r_{3}\right)\left(r_{1}+\hat{B}r_{2}+r_{3}\right)}$
$\displaystyle\Leftrightarrow\quad$