# Optimal consumption-investment with coupled constraints
on consumption and investment strategies
in a regime switching market with random coefficients

Ying Hu Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France. Partially
supported by Lebesgue Center of Mathematics “Investissements d’avenir”program-
ANR-11-LABX-0020-01. Email<EMAIL_ADDRESS>Xiaomin Shi School of
Statistics and Mathematics, Shandong University of Finance and Economics,
Jinan 250100, China. Partially supported by NSFC (No. 11801315), NSF of
Shandong Province (No. ZR2018QA001). Email<EMAIL_ADDRESS>Zuo Quan Xu
Department of Applied Mathematics, The Hong Kong Polytechnic University,
Kowloon, Hong Kong. Partially supported by NSFC (No. 11971409), Hong Kong RGC
(GRF 15202421 and 15204622), The PolyU-SDU Joint Research Center on Financial
Mathematics, The CAS AMSS-PolyU Joint Laboratory of Applied Mathematics, The
Research Centre for Quantitative Finance, The Hong Kong Polytechnic
University. Email<EMAIL_ADDRESS>
This paper studies finite-time optimal consumption-investment problems with
power, logarithmic and exponential utilities, in a regime switching market
with random coefficients, subject to coupled constraints on the consumption
and investment strategies. We provide explicit optimal consumption-investment
strategies and optimal values for the problems in terms of the solutions to
some diagonally quadratic backward stochastic differential equation (BSDE)
systems and linear BSDE systems with unbound coefficients. Some of these BSDEs
are new in the literature and solving them is one of the main theoretical
contributions of this paper. We accomplish the latter by applying the
truncation, approximation technique to get some a priori uniformly lower and
upper bounds for their solutions.

Key words. optimal consumption-investment, regime switching, random
coefficients, backward stochastic differential equation

Mathematics Subject Classification (2020) 91B16 93E20 60H30 91G10

##  1 Introduction

In a consumption-investment model, one aims at maximizing his utility of
consumption and wealth by choosing the best consumption and investment
strategies in a financial market. Following the pioneering work of Merton
[26], a large volume of works has been done on the model. In particular,
various constraints such as bankruptcy prohibition, subsistence consumption
requirement, wealth-dependent investment and consumption constraints are
introduced into the model; see, e.g., Guan, Xu and Yi [9], Sethi [30], Xu and
Yi [35], Zariphopoulou [38]. All these papers focus on Markovian markets. The
model has also been considered in non-Markovian markets with general random
coefficients, by Cox and Huang [5], Karatzas, Lehoczky and Shreve [18],
Matoussi and Xing [25], Xing [34]. The method used in [5] and [18] is known as
martingale method nowadays. Please refer to the seminar monograph Karatzas and
Shreve [19] for a systematic account on this method for more advanced
(incomplete/constrained) markets.

On the other hand, Markov chain is usually adopted to reflect, at the
macroeconomic level, the market status (such as bull and bear) in the
literature. This is the famous regime switching model of market; see Hamilton
[10]. Utility maximization and mean-variance models in a regime switching
market have been studied by many researchers; see, e.g., [2, 24, 32, 31, 33,
37, 39, 40, 36]. In these works, the market parameters are assumed to be
deterministic functions of time for each given regime. This allows the authors
to apply ordinary or partial differential equation (ODE or PDE, for short)
method to solve their problems. In practice, however, even in a bull market,
the market parameters, such as the interest rate, stock appreciation rates and
volatilities are affected by the uncertainties caused by many factors such as
politics, economic, legal, military, corporate governance. Thus, it is too
restrictive to set market parameters as deterministic function of time even if
the market status is known. It is necessary to allow the market parameters to
depend on not only the Markov chain but also other random resources. With this
in mind, the authors consider stochastic linear quadratic control and mean-
variance problems in [13, 14, 15]. Along this framework, this paper
generalizes the consumption-investment model of Cheridito and Hu [4] to a
regime switching market with random coefficients.

This paper studies optimal consumption-investment problems with power,
logarithmic and exponential utilities in a regime switching market with
parameters depending on both a Markov chain and Brownian motion. Because of
the randomness of the market parameters, the ODE and PDE methods will no
longer work for our problem. To deal with the randomness, it is necessary to
apply the backward stochastic differential equation (BSDE) theory of Pardoux
and Peng [28]. Since its inception in 1990, BSDE theory has become a powerful
tool in dealing with modern financial engineering problems such as portfolio
selection and asset pricing problems. For instance, Rouge and El Karoui [29]
characterize the price equation via exponential utility maximization and
quadratic BSDEs. By only requiring the strategies take their values in some
closed sets, Hu, Imkeller and M$\mathrm{\ddot{u}}$ller [11] give solutions to
exponential/power/logarithmic utility maximization of terminal wealth and
Cheridito and Hu [4] take into account the intermediate consumption. Becherer
[3] and Morlais [27] study exponential utility optimization and indifference
valuation with jumps using quadratic BSDEs driven by random measures. Taking
ambiguity and time-consistent ambiguity-averse preferences into account,
Laeven and Stadye [23] investigate the robust portfolio choice and
indifference valuation by quadratic BSDEs with infinite activity jumps.
Kramkov and Pulido [22] solve a quadratic BSDE system arising from a price
impact model. Because of regime switching, the BSDE systems in our problems
are all multidimensional, making them harder to study compared to one
dimensional case.

Meanwhile, in the study of consumption-investment problems, one often assumes
that the consumption and investment strategies are subject to separate
constraints; see, e.g., [4]. In practice, however, the constraints on them may
be coupled together. For instance, when there is no individual constraint on
one’s consumption and investment strategies but he is not allowed to borrow
money, then the constraints are coupled together. This renders the study of
the coupled constraints on the consumption and investment strategies, which
is, to the best of our knowledge, still rare in the literature, partially due
to its complexity. In this paper, different from [4], we consider general, not
necessarily product, coupled constraints on consumption and investment
strategies. This brings new mathematical challenges to our analysis for the
related BSDE systems.

We provide explicit optimal consumption and investment strategies in terms of
solutions to some BSDE systems. Because of the emergence of regime switching,
the BSDE systems in our model are actually diagonally quadratic coupled
through the generator of the Markov chain. Different from [4], the quadratic
BSDE systems in our paper could not be directly covered by existing
literature. Solving these systems is the key theoretical contribution of this
paper and we accomplish this by first doing some approximation and truncation
of the generators, then finding uniformly lower and upper bounds, and
eventually taking limit to obtain the desired solutions. The uniqueness of
their solutions are also proved by pure BSDE techniques, so the method may be
applied other problems in BSDE theory.

The rest part of this paper is organized as follows. At the end of this
section, we introduce some notations and recall some facts about BMO
martingales that will be used frequently in the subsequent analysis. In
Section 2, we present the financial market and formulate the consumption-
investment problem in a regime switching market with random coefficients. In
Sections 3, 4, 5, we solve the problem for the power, logarithmic and
exponential utilities, respectively. For each utility, we first specify the
consumption-investment constraint and define the admissible strategies. We
then accomplish the solvability of some related BSDE systems. With the aid of
their solutions, we finally provide the optimal consumption and investment
strategies and the optimal value of the problem. We conclude the paper in
Section 6.

### Notation

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a fixed complete probability space on
which are defined a standard $n$-dimensional Brownian motion
$W_{t}=(W_{1,t},\ldots,W_{n,t})^{\prime}$ and a continuous-time stationary
Markov chain $\alpha_{t}$ valued in a finite state space
$\mathcal{M}=\\{1,2,\ldots,\ell\\}$ with $\ell\geqslant 1$. We assume
$\\{W_{t}\\}_{t\geqslant 0}$ and $\\{\alpha_{t}\\}_{t\geqslant 0}$ are
independent processes. The Markov chain has a generator
$Q=(q^{ij})_{\ell\times\ell}$ with $q^{ij}\geqslant 0$ for $i\neq j$ and
$\sum_{j=1}^{\ell}q^{ij}=0$ for every $i\in\mathcal{M}$. Define the
filtrations $\mathcal{F}_{t}=\sigma\\{W_{s},\alpha_{s}:0\leqslant s\leqslant
t\\}\bigvee\mathcal{N}$ and $\mathcal{F}^{W}_{t}=\sigma\\{W_{s}:0\leqslant
s\leqslant t\\}\bigvee\mathcal{N}$, where $\mathcal{N}$ is the totality of all
the $\mathbb{P}$-null sets of $\mathcal{F}$.

Throughout this paper, we denote by ${\mathbb{R}}^{n}$ the set of
$n$-dimensional column vectors, by ${\mathbb{R}}^{n}_{+}$ the set of vectors
in ${\mathbb{R}}^{n}$ whose components are nonnegative, by
${\mathbb{R}}^{m\times n}$ the set of $m\times n$ real matrices, and by
$\mathbb{S}^{n}$ the set of symmetric $n\times n$ real matrices. For
$x\in{\mathbb{R}}$, we define $x^{+}:=\max\\{x,0\\}$, and
$x^{-}:=\max\\{-x,0\\}$. If $M=(m_{ij})\in{\mathbb{R}}^{m\times n}$, we denote
its transpose by $M^{\prime}$, and its norm by
$|M|=\sqrt{\sum_{ij}m_{ij}^{2}}$. If $M\in\mathbb{S}^{n}$ is positive definite
(positive semidefinite) , we write $M>$ ($\geqslant$) $0.$ We write $A>$
($\geqslant$) $B$ if $A,B\in\mathbb{S}^{n}$ and $A-B>$ ($\geqslant$) $0.$ We
use the following spaces throughout the paper:

$\displaystyle L^{2}_{\mathcal{F}_{T}}(\Omega;\mathbb{R})$
$\displaystyle=\Big{\\{}\xi:\Omega\rightarrow\mathbb{R}\;\Big{|}\;\xi\mbox{ is
}\mathcal{F}_{T}\mbox{-measurable, and
}{\mathbb{E}}\big{(}|\xi|^{2}\big{)}<\infty\Big{\\}},$ $\displaystyle
L^{\infty}_{\mathcal{F}_{T}}(\Omega;\mathbb{R})$
$\displaystyle=\Big{\\{}\xi:\Omega\rightarrow\mathbb{R}\;\Big{|}\;\xi\mbox{ is
}\mathcal{F}_{T}\mbox{-measurable, and essentially bounded}\Big{\\}},$
$\displaystyle L^{2}_{\mathcal{F}}(0,T;\mathbb{R})$
$\displaystyle=\Big{\\{}\phi:[0,T]\times\Omega\rightarrow\mathbb{R}\;\Big{|}\;\phi\mbox{
is an }\\{\mathcal{F}_{t}\\}_{t\geqslant 0}\mbox{-predictable process with }$
$\displaystyle\qquad\quad\;{\mathbb{E}}\int_{0}^{T}|\phi_{t}|^{2}dt<\infty\Big{\\}},$
$\displaystyle L^{1}_{\mathcal{F}}(0,T;\mathbb{R})$
$\displaystyle=\Big{\\{}\phi:[0,T]\times\Omega\rightarrow\mathbb{R}\;\Big{|}\;\phi\mbox{
is an }\\{\mathcal{F}_{t}\\}_{t\geqslant 0}\mbox{-predictable process with }$
$\displaystyle\qquad\quad\;{\mathbb{E}}\int_{0}^{T}|\phi_{t}|dt<\infty\Big{\\}},$
$\displaystyle L^{\infty}_{\mathcal{F}}(0,T;\mathbb{R})$
$\displaystyle=\Big{\\{}\phi:[0,T]\times\Omega\rightarrow\mathbb{R}\;\Big{|}\;\phi\mbox{
is an }\\{\mathcal{F}_{t}\\}_{t\geqslant 0}\mbox{-predictable essentially}$
$\displaystyle\qquad\mbox{\quad\ bounded process}\Big{\\}}.$

These definitions are generalized in the obvious way to the cases that
$\mathcal{F}$ is replaced by $\mathcal{F}^{W}$ and $\mathbb{R}$ by
$\mathbb{R}^{n}$, $\mathbb{R}^{n\times m}$ or $\mathbb{S}^{m}$.

In our analysis, some arguments such $s$, $t$, $\omega$, as well as some terms
including “almost surely” (a.s.) and “almost everywhere” (a.e.) may be
suppressed for notation simplicity in some circumstances when no confusion
occurs.

### BMO martingales

We recall some facts about BMO martingales (see Kazamaki [20]). They will be
used in our subsequent analysis.

For a $\Lambda\in L^{2}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$, the process
$\int_{0}^{\cdot}\Lambda_{s}^{\prime}dW_{s}$ is called a BMO martingale (on
$[0,T]$) if there exists a constant $K>0$ such that

$\mathbb{E}\bigg{[}\int_{\tau}^{T}|\Lambda_{s}|^{2}ds\;\bigg{|}\;\mathcal{F}_{\tau}^{W}\bigg{]}\leqslant
K$

for all $\\{\mathcal{F}_{t}^{W}\\}_{t\geqslant 0}$-stopping times
$\tau\leqslant T$. The set of BMO martingales is defined as

$\displaystyle L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$
$\displaystyle=\bigg{\\{}\Lambda\in
L^{2}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})\;\bigg{|}\;\int_{0}^{\cdot}\Lambda_{s}^{\prime}dW_{s}\mbox{
is a BMO martingale on $[0,T]$}\bigg{\\}}.$

The following two important properties of BMO martingales will be used in our
below arguments without claim.

  * •

If $\int_{0}^{\cdot}\Lambda_{s}^{\prime}dW_{s}$ is a BMO martingale, then
Dol$\acute{\mathrm{e}}$ans-Dade stochastic exponential

$\mathcal{E}\bigg{(}\int_{0}^{\cdot}\Lambda_{s}^{\prime}dW_{s}\bigg{)}$

is a uniformly integrable martingale.

  * •

If $\int_{0}^{\cdot}\Lambda_{s}^{\prime}dW_{s}$ and
$\int_{0}^{\cdot}Z_{s}^{\prime}dW_{s}$ are both BMO martingales, then
$\widetilde{W}:=W-\int_{0}^{\cdot}Z_{s}ds$ is a standard Brownian motion, and
$\int_{0}^{\cdot}\Lambda_{s}^{\prime}d\widetilde{W}_{s}$ is a BMO martingale
under the probability measure $\widetilde{\mathbb{P}}$ defined by

$\frac{d\widetilde{\mathbb{P}}}{d\mathbb{P}}\bigg{|}_{\mathcal{F}^{W}_{T}}=\mathcal{E}\bigg{(}\int_{0}^{T}Z_{s}^{\prime}dW_{s}\bigg{)}.$

##  2 Problem formulation

###  2.1 Financial market

Consider a financial market consisting of a risk-free asset (the money market
instrument or bond) whose price is $S_{0}$ and $m$ risky securities (the
stocks) whose prices are $S_{1},\ldots,S_{m}$. Assume $m\leqslant n$, i.e.,
the number of risky securities is no more than that of the market risk
resources (namely the Brownian motion). When $m<n$ the market is incomplete.
These asset prices are driven by SDEs:

$\displaystyle\begin{cases}dS_{0,t}=r_{t}^{\alpha_{t}}S_{0,t}dt,\\\
S_{0,0}=s_{0},\end{cases}$

and

$\displaystyle\begin{cases}dS_{k,t}=S_{k,t}\Big{(}\mu_{k,t}^{\alpha_{t}}dt+\sum\limits_{j=1}^{n}\sigma_{kj,t}^{\alpha_{t}}dW_{j,t}\Big{)},\\\
S_{k,0}=s_{k},\end{cases}$

where $r^{i}_{t}$ is the interest rate process and $\mu_{k,t}^{i}$ and
$\sigma_{k,t}^{i}:=(\sigma_{k1,t}^{i},\ldots,\sigma_{kn,t}^{i})$ are the
appreciation rate process and volatility rate process of the $k$th risky
security corresponding to a market regime $\alpha_{t}=i$, for every
$k=1,\ldots,m$ and $i\in\mathcal{M}$. Recall that $\alpha_{t}$ follows a
continuous-time stationary Markov chain valued in a finite state space
$\mathcal{M}=\\{1,2,\ldots,\ell\\}$ with $\ell\geqslant 1$. When $\ell=1$,
there is no regime switching and the market becomes the classical Black-
Scholes market. In our below argument we assume $\ell>1$, although all the
results remain true if $\ell=1$.

Define the appreciate vector

$\displaystyle\mu_{t}^{i}=(\mu_{1,t}^{i},\ldots,\mu_{m,t}^{i})^{\prime},$

and volatility matrix

$\displaystyle\sigma_{t}^{i}=\left(\begin{array}[]{c}\sigma_{1,t}^{i}\\\
\vdots\\\ \sigma_{m,t}^{i}\\\
\end{array}\right)\equiv(\sigma_{kj,t}^{i})_{m\times n},\ \text{for}\
\text{each}\quad i\in\mathcal{M}.$

###  2.2 Optimal investment-consumption problem

Consider a small investor, whose actions cannot affect the asset prices. He
will decide at every time $t\in[0,T]$ the _proportion_ $\pi_{j,t}$ of his
wealth to invest in the $j$th risky asset, $j=1,\ldots,m$, as well as the
_proportion_ $c_{t}$ of his wealth to consume. The vector process
$\pi_{t}:=(\pi_{1,t},\ldots,\pi_{m,t})^{\prime}$ is called a portfolio of the
investor. The pair $(\pi,c)$ is called a consumption-investment strategy. The
investor’s self-financing wealth process $X$ corresponding to a consumption-
investment strategy $(\pi,c)$ is a strong solution of the SDE (see, e.g.,
[19]):

$\displaystyle\begin{cases}dX_{t}=X_{t}[r_{t}^{\alpha_{t}}+\pi_{t}^{\prime}b_{t}^{\alpha_{t}}-c_{t}]dt+X_{t}\pi_{t}^{\prime}\sigma_{t}^{\alpha_{t}}dW_{t},\\\
X_{0}=x>0,\ \alpha_{0}=i_{0}\in\mathcal{M},\end{cases}$ (2.1)

where
$b_{t}^{\alpha_{t}}:=\mu_{t}^{\alpha_{t}}-r_{t}^{\alpha_{t}}\mathbf{1}_{m}$
and $\mathbf{1}_{m}$ is the $m$-dimensional vector with all entries being one.
By Itô’s lemma, one can easily show that the process $X$ is always positive,
hence the investor would never be bankrupt.

The investor’s problem is to maximize

$\displaystyle J(x,i_{0};\pi,c)$
$\displaystyle:={\mathbb{E}}\bigg{[}\int_{0}^{T}e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}U(c_{t}X_{t})dt+e^{-\int_{0}^{T}\rho_{s}^{\alpha_{s}}ds}U(X_{T})\bigg{]},\quad\mathrm{s.t.}\;\;(\pi,c)\in\mathcal{U},$
(2.2)

and determine the value function

$V(x,i_{0}):=\sup_{(\pi,c)\in\mathcal{U}}J(x,i_{0};\pi,c),$

where $\rho^{i}\in L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})$,
$i\in\mathcal{M}$, are the discount factor processes, $U$ is the utility
function of the investor, and $\mathcal{U}$ is the admissible set of
consumption-investment strategies. The utility $U$ and admissible set
$\mathcal{U}$ will be defined in the sequel case by case. In particular, the
consumption-investment strategies can be subject to fairly general
constraints. Let $\Theta$ denote the constraint set for them, which is assumed
to be a given closed nonempty set in $\mathbb{R}^{m}\times\mathbb{R}_{+}$.

###### Example 2.1

Here are some important and interesting examples for the constraint set
$\Theta$ in financial practice.

  * •

If the investor are not allowed to borrow money, then the total consumption
and investment cannot beyond 100%, that is,

$\displaystyle\Theta$
$\displaystyle=\Big{\\{}(\pi,c)\in\mathbb{R}^{m}\times\mathbb{R}_{+}\;\Big{|}\;\sum_{j=1}^{m}\pi_{j}+c\leqslant
1\Big{\\}}.$

Because the constraints on the consumption and investment strategies are
coupled together, this will bring some mathematical challenges to our analysis
for the related BSDE systems below.

  * •

If no shorting is allowed in the stocks, then

$\displaystyle\Theta$
$\displaystyle=\Big{\\{}(\pi,c)\in\mathbb{R}^{m}\times\mathbb{R}_{+}\;\Big{|}\;\pi_{j}\geqslant
0,\;j=1,\ldots,m\Big{\\}}.$

In this case $\Theta$ is cone.

  * •

There may be restrictions on the investment strategies on some stocks; for
instance

$\displaystyle\Theta$
$\displaystyle=\Big{\\{}(\pi,c)\in\mathbb{R}^{m}\times\mathbb{R}_{+}\;\Big{|}\;\pi_{j}\in[d_{j},e_{j}],\;j=1,\ldots,m\Big{\\}};$

or constraint on the consumption strategies such as

$\displaystyle\Theta$
$\displaystyle=\Big{\\{}(\pi,c)\in\mathbb{R}^{m}\times\mathbb{R}_{+}\;\Big{|}\;c\in[0,\tfrac{1}{5}]\Big{\\}};$

or coupled constraints on both the investment and consumption strategies

$\displaystyle\Theta$
$\displaystyle=\Big{\\{}(\pi,c)\in\mathbb{R}^{m}\times\mathbb{R}_{+}\;\Big{|}\;\;c\in[0,\tfrac{1}{5}],\;\pi_{j}\in[d_{j},e_{j}],\;j=1,\ldots,m\Big{\\}}.$

We put the following standard assumption for the market parameters.

###### Assumption 1

For all $i\in\mathcal{M}$,

$\displaystyle\begin{cases}r^{i}\in
L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}),\ \mu^{i}\in
L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}^{m}),\\\ \sigma^{i}\in
L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}^{m\times n}),\ \rho^{i}\in
L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}).\end{cases}$

Also, there exists a constant $\delta>0$ such that
$\sigma^{i}(\sigma^{i})^{\prime}\geqslant\delta I_{m}$ for all
$i\in\mathcal{M}$, where $I_{m}$ denotes the $m$-dimensional identity matrix.

Due to different features of the power, logarithmic and exponential utilities,
they must be dealt with different methods. We study them in the subsequent
Sections 3, 4 and 5, respectively.

##  3 Power utility

In this section, we assume that the investor’s utility function is

$U(x)=\frac{1}{\gamma}x^{\gamma},\quad x>0,\quad\gamma\in(-\infty,0)\cup(0,1)$

and the admissible consumption-investment set is defined as

$\displaystyle\mathcal{U}=\bigg{\\{}(\pi,c)\;\bigg{|}\;\int_{0}^{T}(|\pi_{t}|^{2}+c_{t})dt<\infty,a.s.,\
(\pi_{t}(\omega),c_{t}(\omega))\in\Theta,\ a.e.,\ a.s,\bigg{\\}},$

if $\gamma\in(0,1)$; and

$\displaystyle\mathcal{U}$
$\displaystyle=\bigg{\\{}(\pi,c)\;\bigg{|}\;\int_{0}^{T}(|\pi_{t}|^{2}+c_{t})dt<\infty,a.s.,\
(\pi_{t}(\omega),c_{t}(\omega))\in\Theta,\ a.e.,\ a.s,$
$\displaystyle\qquad\qquad\qquad(X_{t}^{\gamma})_{0\leqslant t\leqslant T}\
\mbox{belongs to class (D) and }\
{\mathbb{E}}\bigg{[}\int_{0}^{T}(c_{t}X_{t})^{\gamma}dt\bigg{]}<\infty\bigg{\\}},$

if $\gamma<0$.111We say $(X_{t}^{\gamma})_{0\leqslant t\leqslant T}$ belongs
to class (D) if $\\{X_{\tau}^{\gamma}:\tau\ \mbox{stopping time valued in }\
[0,T]\\}$ is a family of uniformly integrable random variables.

###### Assumption 2

If $\gamma\in(0,1)$, then no consumption or investment is always permitted,
namely $(0\mathbf{1}_{m},0)\in\Theta$. If $\gamma<0$, then positive
consumption is always needed (for otherwise the admissible set $\mathcal{U}$
ie empty), there exists $\varepsilon>0$ such that
$(0\mathbf{1}_{m},\varepsilon)\in\Theta$.

To tackle problem (2.2), we introduce the following $\ell$-dimensional BSDE
system:

$\displaystyle\begin{cases}dP_{t}^{i}=-[f^{i}(P_{t}^{i},\Lambda_{t}^{i})-(\rho^{i}-\gamma
r^{i})P^{i}+\sum_{j=1}^{\ell}q^{ij}P^{j}]dt+(\Lambda_{t}^{i})^{\prime}dW,\\\
P_{T}^{i}=1,\\\ P^{i}>0,\quad i\in\mathcal{M},\end{cases}$ (3.1)

where, for any $(P,\Lambda)\in\mathbb{R}_{+}\times\mathbb{R}^{n}$,

$\displaystyle f^{i}(P,\Lambda)$
$\displaystyle=\gamma\operatorname*{ess\;sup}_{(\pi,c)\in\Theta}\bigg{[}-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{i}|^{2}P+\pi^{\prime}(Pb^{i}+\sigma^{i}\Lambda)+\frac{c^{\gamma}}{\gamma}-Pc\bigg{]}.$

###### Remark 3.1

If there is no consumption-investment constraint, namely
$\Theta=\mathbb{R}^{n}\times\mathbb{R}_{+}$, then

$\displaystyle f^{i}(P,\Lambda)$
$\displaystyle=\frac{1}{2}\frac{\gamma}{1-\gamma}\frac{1}{P}(Pb+\sigma\Lambda)^{\prime}(\sigma\sigma^{\prime})^{-1}(Pb+\sigma\Lambda)+(1-\gamma)P^{-\frac{\gamma}{1-\gamma}}.$

###### Definition 3.2

A vector process $(P^{i},\Lambda^{i})_{i\in\mathcal{M}}$ is called a solution
to the $\ell$-dimensional BSDE (3.1), if it satisfies (3.1), and
$(P^{i},\Lambda^{i})\in L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ for all
$i\in\mathcal{M}$. A solution $(P^{i},\Lambda^{i})_{i\in\mathcal{M}}$ to (3.1)
is called uniformly positive if $P^{i}_{t}\geqslant\delta$, for all
$t\in[0,T]$, a.s. with some deterministic constant $\delta>0$.

The following comparison theorem for multidimensional BSDE systems firstly
appeared in [16] (one can find a concise version in [12, Lemma 2.2] or [13,
Lemma 3.4]). We shall use it frequently in the study of BSDEs.

###### Lemma 3.3

Suppose $(Y^{i},Z^{i})_{i\in\mathcal{M}}$ and
$(\overline{Y}^{i},\overline{Z}^{i})_{i\in\mathcal{M}}$ satisfy the following
two $\ell$-dimensional BSDE systems, respectively:

$\displaystyle
Y^{i}_{t}=\xi^{i}+\int_{t}^{T}g^{i}(s,Y^{i}_{s},Y^{-i}_{s},Z^{i}_{s})ds-\int_{t}^{T}(Z^{i}_{s})^{\prime}dW_{s},\
\mbox{ for all $i\in\mathcal{M}$;}$

and

$\displaystyle\overline{Y}^{i}_{t}=\overline{\xi}^{i}+\int_{t}^{T}\overline{g}^{i}(s,\overline{Y}^{i}_{s},\overline{Y}^{-i}_{s},\overline{Z}^{i}_{s})ds-\int_{t}^{T}(\overline{Z}^{i}_{s})^{\prime}dW_{s},\
\mbox{ for all $i\in\mathcal{M}$, }$

where
$Y^{-i}_{s}=(Y^{1}_{s},\ldots,Y^{i-1}_{s},Y^{i+1}_{s},\ldots,Y^{\ell}_{s})$.
Also suppose that, for all $i\in\mathcal{M}$,

  1. 1.

$\xi^{i},\ \overline{\xi}^{i}\in L^{2}_{\mathcal{F}^{W}}(\Omega;\mathbb{R})$,
and $\xi^{i}\leqslant\overline{\xi}^{i}$;

  2. 2.

there exists a constant $K>0$ such that

$|g^{i}(s,y,z)-g^{i}(s,\overline{y},\overline{z})|\leqslant
K(|y-\overline{y}|+|z-\overline{z}|),$

for any $z,\overline{z}\in\mathbb{R}^{n}$, $y=(y^{i},y^{-i})$,
$\overline{y}=(\overline{y}^{i},\overline{y}^{-i})\in\mathbb{R}^{\ell}$;

  3. 3.

$g^{i}(s,y,z)$ is nondecreasing in every $y^{j}$, $j\neq i\in\mathcal{M}$; and

  4. 4.

$g^{i}(s,\overline{Y}^{i}_{s},\overline{Y}^{-i}_{s},\overline{Z}^{i}_{s})\leqslant\overline{g}^{i}(s,\overline{Y}^{i}_{s},\overline{Y}^{-i}_{s},\overline{Z}^{i}_{s})$.

Then $Y^{i}_{t}\leqslant\overline{Y}^{i}_{t}$ a.s. for all $t\in[0,T]$ and
$i\in\mathcal{M}$.

###### Remark 3.4

If conditions 2 and 3 hold for $\overline{g}^{i}$ instead of $g^{i}$, and
condition 4 is replaced by

$g^{i}(s,Y^{i}_{s},Y^{-i}_{s},Z^{i}_{s})\leqslant\overline{g}^{i}(s,Y^{i}_{s},Y^{-i}_{s},Z^{i}_{s})$

in the above lemma, then the conclusion still holds.

###### Theorem 3.5

Suppose $\gamma\in(0,1)$ and Assumptions 1 and 2 hold. Then there is a unique
uniformly positive solution $(P^{i},\Lambda^{i})_{i\in\mathcal{M}}$ to the
BSDE (3.1).

Proof: Let $a>1$ be a large constant such that

$\displaystyle-\rho^{i}+\gamma r^{i}\geqslant-a,$ (3.2)

and

$\displaystyle\frac{\gamma}{2(1-\gamma)}(b^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}b^{i}-\rho^{i}+\gamma
r^{i}\leqslant a(1-\gamma),$ (3.3)

hold simultaneously for all $i\in\mathcal{M}$. Denote $a_{1}:=e^{-aT}$ and
$a_{2}:=2e^{aT}$.

Let $g:{\mathbb{R}}\rightarrow[0,1]$ be a smooth truncation function
satisfying $g(x)=0$ for $x\in(-\infty,\frac{1}{2}a_{1}]$, and $g(x)=1$ for
$x\in[a_{1},+\infty)$. For $k\geqslant 1$,
$(t,P,\Lambda)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{n}$,
$i\in\mathcal{M}$, define

$f^{k,i}(t,P,\Lambda)=\operatorname*{ess\;inf}_{\tilde{P}\in\mathbb{R},\tilde{\Lambda}\in\mathbb{R}^{n}}\Big{[}f^{i}(t,\tilde{P},\tilde{\Lambda})g(\tilde{P})+k|P-\tilde{P}|+k|\Lambda-\tilde{\Lambda}|\Big{]}.$

Then it is uniformly Lipschitz continuous in $(P,\Lambda)$, and increasingly
approaches to $f^{i}(t,P,\Lambda)g(P)$ as $k$ goes to infinity. According to
Assumption 2, we have $f^{i}\geqslant 0$, hence $f^{k,i}\geqslant 0$.

The following BSDE system

$\displaystyle\begin{cases}dP^{k,i}=-\Big{[}f^{k,i}(P^{k,i},\Lambda^{k,i})-(\rho^{i}-\gamma
r^{i})P^{k,i}+\sum_{j=1}^{\ell}q^{ij}P^{k,j}\Big{]}dt+(\Lambda^{k,i})^{\top}dW,\\\
P^{k,i}_{T}=1,\ \mbox{ for all $i\in\mathcal{M}$, }\end{cases}$

has a Lipschitz generator222As for BSDE
$Y_{t}=\xi+\int_{t}^{T}f(s,Y_{s},Z_{s})ds-\int_{t}^{T}Z_{s}^{\prime}dW_{s}$,
$f$ is called the generator and $\xi$ is called the terminal value., so it
admits a unique solution, denoted by
$\big{(}P^{k,i},\Lambda^{k,i}\big{)}_{i\in\mathcal{M}}$.

It is direct to verify that

$\displaystyle(\underline{P}^{i}_{t},\underline{\Lambda}^{i}_{t})=\Big{(}e^{-a(T-t)},0\Big{)},\quad
i\in\mathcal{M}$

is the unique solution to the following linear BSDE system:

$\displaystyle\begin{cases}d\underline{P}^{i}=-\Big{[}-a\underline{P}^{i}+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}\Big{]}dt+(\underline{\Lambda}^{i})^{\prime}dW,\\\
\underline{P}^{i}_{T}=1,\quad i\in\mathcal{M}.\end{cases}$

Notice that $f^{k,i}(P,\Lambda)\geqslant 0$ and recall (3.2), so

$\displaystyle
f^{k,i}(\underline{P}^{i},\underline{\Lambda}^{i})-(\rho^{i}-\gamma
r^{i})\underline{P}^{i}+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}$
$\displaystyle\geqslant-(\rho^{i}-\gamma
r^{i})\underline{P}^{i}+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}\geqslant-a\underline{P}^{i}+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}.$

By Lemma 3.3, we have

$a_{1}\leqslant e^{-a(T-t)}=\underline{P}^{i}_{t}\leqslant P^{k,i}_{t},\quad
i\in\mathcal{M}.$

On the other hand,

$\displaystyle(\bar{P}^{i},\bar{\Lambda}^{i})=\bigg{(}\Big{(}e^{a(T-t)}+\frac{1}{a}(e^{a(T-t)}-1)\Big{)}^{1-\gamma},\;0\bigg{)},\quad
i\in\mathcal{M},$

is a solution to the following BSDE system:

$\displaystyle\begin{cases}d\bar{P}^{i}=-\Big{[}a(1-\gamma)\bar{P}^{i}+(1-\gamma)(\bar{P}^{i})^{-\frac{\gamma}{1-\gamma}}+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}\Big{]}dt+(\bar{\Lambda}^{i})^{\prime}dW,\\\
\bar{P}^{i}_{T}=1,\quad i\in\mathcal{M}.\end{cases}$

Notice, for $P>0$,

$\displaystyle f^{k,i}(t,P,\Lambda)$ $\displaystyle\leqslant
f^{i}(t,P,\Lambda)g(P)$
$\displaystyle\leqslant\gamma\operatorname*{ess\;sup}_{(\pi,c)\in{\mathbb{R}}^{n}\times{\mathbb{R}}_{+}}\Big{[}-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{i}|^{2}P+\pi^{\prime}(Pb^{i}+\sigma^{i}\Lambda)+\frac{c^{\gamma}}{\gamma}-Pc\Big{]}g(P)$
$\displaystyle=\frac{1}{2}\frac{\gamma}{1-\gamma}\frac{1}{P}(Pb+\sigma\Lambda)^{\prime}(\sigma\sigma^{\prime})^{-1}(Pb+\sigma\Lambda)g(P)+(1-\gamma)P^{-\frac{\gamma}{1-\gamma}}g(P)$
$\displaystyle\leqslant\frac{\gamma}{2(1-\gamma)}(b^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}b^{i}P+\frac{\gamma}{1-\gamma}(b^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}\sigma^{i}\Lambda$
$\displaystyle\qquad+\frac{\gamma}{2(1-\gamma)}\frac{1}{P}(\sigma^{i}\Lambda)^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}\sigma^{i}\Lambda+(1-\gamma)P^{-\frac{\gamma}{1-\gamma}}.$
(3.4)

Hence, by (3.3),

$\displaystyle\quad f^{k,i}(\bar{P}^{i},\bar{\Lambda}^{i})-(\rho^{i}-\gamma
r^{i})\bar{P}^{i}+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}$
$\displaystyle\leqslant\frac{\gamma}{2(1-\gamma)}(b^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}b^{i}\bar{P}^{i}+(1-\gamma)(\bar{P}^{i})^{-\frac{\gamma}{1-\gamma}}-(\rho^{i}-\gamma
r^{i})\bar{P}^{i}+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}$ $\displaystyle\leqslant
a(1-\gamma)\bar{P}^{i}+(1-\gamma)(\bar{P}^{i})^{-\frac{\gamma}{1-\gamma}}+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}.$

Then by Lemma 3.3 again, we have

$P^{k,i}_{t}\leqslant\bar{P}^{i}_{t}\leqslant 2e^{aT}=a_{2},$

and $P^{k,i}$ is increasing in $k$, for each $i\in\mathcal{M}$.

Let $P^{i}_{t}=\lim\limits_{k\rightarrow\infty}P^{k,i}_{t}$,
$i\in\mathcal{M}$. Since $a_{1}$ and $a_{2}$ are independent of $k$,
$a_{1}\leqslant P^{i}_{t}\leqslant a_{2}$. Recalling $f^{k,i}\geqslant 0$, (3)
and the role of the truncation function $g$, we can regard
$\big{(}P^{k,i},\Lambda^{k,i}\big{)}$ as the solution of a scalar-valued
quadratic BSDE for each $i\in\mathcal{M}$. Thus by [21, Proposition 2.4],
there exists a process $\Lambda\in
L^{2}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n\times\ell})$ such that $(P,\Lambda)$
satisfies the BSDE (3.1).

Applying Itô’s formula to $(P^{i}-a_{2})^{2}$, we obtain, for any stopping
times $\tau\leqslant T$,

$\displaystyle{\mathbb{E}}\bigg{[}\int_{\tau}^{T}|\Lambda^{i}|^{2}\;\bigg{|}\;\mathcal{F}_{\tau}^{W}\bigg{]}$
$\displaystyle=(1-a_{2})^{2}-(P_{\tau}^{i}-a_{2})^{2}$
$\displaystyle\qquad+2{\mathbb{E}}\bigg{[}\int_{\tau}^{T}(P^{i}-a_{2})\Big{[}f^{i}(P^{i},\Lambda^{i})-(\rho^{i}-\gamma
r^{i})P^{i}+\sum_{j=1}^{\ell}q^{ij}P^{j}\Big{]}ds\;\bigg{|}\;\mathcal{F}_{\tau}^{W}\bigg{]}$
$\displaystyle\leqslant(1-a_{2})^{2}-(P_{\tau}^{i}-a_{2})^{2}$
$\displaystyle\qquad+2{\mathbb{E}}\bigg{[}\int_{\tau}^{T}(P^{i}-a_{2})\Big{[}-(\rho^{i}-\gamma
r^{i})P^{i}+\sum_{j=1}^{\ell}q^{ij}P^{j}\Big{]}ds\;\bigg{|}\;\mathcal{F}_{\tau}^{W}\bigg{]}$
$\displaystyle\leqslant(1-a_{2})^{2}+2{\mathbb{E}}\bigg{[}\int_{0}^{T}|P^{i}-a_{2}|\Big{|}-(\rho^{i}-\gamma
r^{i})P^{i}+\sum_{j=1}^{\ell}q^{ij}P^{j}\Big{|}ds\;\bigg{|}\;\mathcal{F}_{\tau}^{W}\bigg{]},$

where we used the fact that $P^{i}\leqslant a_{2}$ and $f^{i}\geqslant 0$ to
get the first inequity. Because $a_{1}\leqslant P^{i}\leqslant a_{2}$ and
Assumption 1, we see the right hand side is upper bounded by a constant. Hence
$\Lambda^{i}\in L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$,
for all $i\in\mathcal{M}$. We have now established the existence of the
solution.

Next, let us prove the uniqueness. Suppose $(P^{i},\
\Lambda^{i})_{i\in\mathcal{M}}$, $(\tilde{P}^{i},\
\tilde{\Lambda}^{i})_{i\in\mathcal{M}}$ are two uniformly positive solutions
of $\eqref{P1}$. For every $i\in\mathcal{M}$, define processes

$\displaystyle(Y_{t}^{i},Z_{t}^{i})=\left(\ln
P_{t}^{i},\;\frac{\Lambda_{t}^{i}}{P_{t}^{i}}\right),\
(\tilde{Y}_{t}^{i},\tilde{Z}_{t}^{i})=\left(\ln\tilde{P}_{t}^{i},\;\frac{\tilde{\Lambda}_{t}^{i}}{\tilde{P}_{t}^{i}}\right),\
\mbox{ for }t\in[0,T].$ (3.5)

Then $(Y^{i},Z^{i})$, $(\tilde{Y}^{i},\tilde{Z}^{i})\in
L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$, for all
$i\in\mathcal{M}$. Furthermore, by Itô’s formula,
$(Y^{i},Z^{i})_{i\in\mathcal{M}}$ and
$(\tilde{Y}^{i},\tilde{Z}^{i})_{i\in\mathcal{M}}$ satisfy the following BSDE
system:

$\displaystyle\begin{cases}dY^{i}=-\Big{[}F^{i}(Y^{i},Z^{i})+\frac{1}{2}|Z^{i}|^{2}-\rho^{i}+\gamma
r^{i}+\sum_{j=1}^{\ell}q^{ij}e^{Y^{j}-Y^{i}}\Big{]}dt+(Z^{i})^{\prime}dW,\\\
Y^{i}_{T}=0,\quad i\in\mathcal{M}.\end{cases}$ (3.6)

where for any $(Y,Z)\in\mathbb{R}\times\mathbb{R}^{n}$,

$\displaystyle F^{i}(Y,Z)$
$\displaystyle=\gamma\operatorname*{ess\;sup}\limits_{(\pi,c)\in\Theta}\Big{[}-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(b^{i}+\sigma^{i}Z)+\frac{c^{\gamma}}{\gamma}e^{-Y}-c\Big{]}.$

Because $\ln a_{1}\leqslant Y^{i}\leqslant\ln a_{2}$ and use Assumption 1, for
any $(\pi,c)\in\Theta$,

$\displaystyle\quad-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(b^{i}+\sigma^{i}Z^{i})+\frac{c^{\gamma}}{\gamma}e^{-Y^{i}}-c$
$\displaystyle\leqslant-\delta\frac{1-\gamma}{2}|\pi|^{2}+K_{1}|\pi|(1+|Z|)+\sup_{c\in{\mathbb{R}}_{+}}\Big{(}\frac{c^{\gamma}}{\gamma}\frac{1}{a_{1}}-c\Big{)}$
$\displaystyle=-\delta\frac{1-\gamma}{2}|\pi|^{2}+K_{1}|\pi|(1+|Z|)+\frac{1-\gamma}{\gamma}a_{1}^{\frac{1}{1-\gamma}}$
$\displaystyle<0,$

if $|\pi|>K(1+|Z^{i}|)$ for sufficient large $K$.333 Hereafter, we shall use
$K$ to represent a generic positive constant independent of $i,\ m,\ n$ and
$t$, which can be different from line to line. Since
$0\mathbf{1}_{m+1}\in\Theta$, $F^{i}(Y,Z)\geqslant 0$. Therefore

$\displaystyle F^{i}(Y^{i},Z^{i})$
$\displaystyle=\gamma\operatorname*{ess\;sup}_{\begin{subarray}{c}(\pi,c)\in\Theta\\\
|\pi|\leqslant
K(1+|Z^{i}|)\end{subarray}}\Big{[}-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(b^{i}+\sigma^{i}Z^{i})+\frac{c^{\gamma}}{\gamma}e^{-Y^{i}}-c\Big{]}.$
(3.7)

This still holds if we replace $K(1+|Z^{i}|)$ by any bigger value. A similar
expression holds for $F^{i}(\tilde{Y}^{i},\tilde{Z}^{i})$.

Set $\overline{Y}^{i}=Y^{i}-\tilde{Y}^{i},\
\overline{Z}^{i}=Z^{i}-\tilde{Z}^{i}$, for $i\in\mathcal{M}$. From (3.7), we
have

$\displaystyle\Big{|}F^{i}(Y^{i},Z^{i})-F^{i}(Y^{i},\tilde{Z}^{i})\Big{|}$
$\displaystyle\leqslant\gamma\operatorname*{ess\;sup}_{\begin{subarray}{c}(\pi,c)\in\Theta\\\
|\pi|\leqslant
K(1+|Z^{i}|+|\tilde{Z}^{i}|)\end{subarray}}|\pi^{\prime}\sigma^{i}\overline{Z}^{i}|\leqslant
K(1+|Z^{i}|+|\tilde{Z}^{i}|)|\overline{Z}^{i}|.$

Since $Z^{i},\tilde{Z}^{i}\in
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$, we can define
$\beta^{i}\in L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ in
an obvious way such that

$F^{i}(Y^{i},Z^{i})-F^{i}(Y^{i},\tilde{Z}^{i})=(\beta^{i})^{\prime}\overline{Z}^{i},$

and

$|\beta^{i}|\leqslant K(1+|Z^{i}|+|\tilde{Z}^{i}|).$

Now applying Itô’s formula to $(\overline{Y}^{i})^{2}$, we deduce that

$\displaystyle(\overline{Y}^{i}_{t})^{2}$
$\displaystyle=\int_{t}^{T}\Big{\\{}2\overline{Y}^{i}\Big{[}F^{i}(Y^{i},Z^{i})-F^{i}(\tilde{Y}^{i},\tilde{Z}^{i})+\frac{1}{2}(|Z^{i}|^{2}-|\tilde{Z}^{i}|^{2})$
$\displaystyle\qquad+\sum_{j=1}^{\ell}q^{ij}\Big{(}e^{Y^{j}-Y^{i}}-e^{\tilde{Y}^{j}-\tilde{Y}^{i}}\Big{)}\Big{]}-|\overline{Z}^{i}|^{2}\Big{\\}}ds-\int_{t}^{T}2\overline{Y}^{i}(\overline{Z}^{i})^{\prime}dW.$
(3.8)

Notice that the map

$Y\mapsto F^{i}(Y,Z),\ Y\in{\mathbb{R}}_{+},$

is non-increasing for every $Z\in{\mathbb{R}}^{n}$ and $i\in\mathcal{M}$.
Therefore,

$\displaystyle\overline{Y}^{i}\big{[}F^{i}(Y^{i},Z^{i})-F^{i}(\tilde{Y}^{i},\tilde{Z}^{i})\big{]}$
$\displaystyle=\overline{Y}^{i}\big{[}F^{i}(Y^{i},Z^{i})-F^{i}(Y^{i},\tilde{Z}^{i})+F^{i}(Y^{i},\tilde{Z}^{i})-F^{i}(\tilde{Y}^{i},\tilde{Z}^{i})\big{]}$
$\displaystyle\leqslant\overline{Y}^{i}\big{[}F^{i}(Y^{i},Z^{i})-F^{i}(Y^{i},\tilde{Z}^{i})\big{]}$
$\displaystyle=\overline{Y}^{i}(\beta^{i})^{\prime}\overline{Z}^{i}.$

For each fixed $i\in\mathcal{M}$, let us introduce the process

$N^{i}_{t}=\mathcal{E}\Big{(}\int_{0}^{t}(\beta^{i}_{s}+\frac{1}{2}(Z^{i}+\tilde{Z}^{i}))^{\prime}dW_{s}\Big{)}.$

Then $N_{t}^{i}$ is a uniformly integrable martingale. Notice $Y^{i}$,
$i\in\mathcal{M}$ are bounded, from (3),

$\displaystyle(\overline{Y}^{i}_{t})^{2}$
$\displaystyle\leqslant\int_{t}^{T}\Big{\\{}2\overline{Y}^{i}\Big{[}(\beta^{i})^{\prime}\overline{Z}^{i}+\frac{1}{2}(Z^{i}+\tilde{Z}^{i})^{\prime}\overline{Z}^{i}+K\sum_{j\neq
i}^{\ell}\overline{Y}^{j}\Big{]}\Big{\\}}ds-\int_{t}^{T}2\overline{Y}^{i}(\overline{Z}^{i})^{\prime}dW$
$\displaystyle=\int_{t}^{T}2K\overline{Y}^{i}\sum_{j\neq
i}^{\ell}\overline{Y}^{j}ds-\int_{t}^{T}2\overline{Y}^{i}(\overline{Z}^{i})^{\prime}d\widetilde{W}^{i}$

where

$\widetilde{W}^{i}_{t}:=W_{t}-\int_{0}^{t}\Big{(}\beta^{i}_{s}+\frac{1}{2}(Z^{i}+\tilde{Z}^{i})\Big{)}ds,$

is a Brownian motion under the probability $\widetilde{\mathbb{P}}^{i}$
defined by

$\displaystyle\frac{d\widetilde{\mathbb{P}^{i}}}{d\mathbb{P}}\bigg{|}_{\mathcal{F}_{T}^{W}}=N^{i}_{T}.$

Taking expectation $\widetilde{\mathbb{E}}^{i}$ w.r.t. the probability measure
$\widetilde{\mathbb{P}}^{i}$,

$\displaystyle(\overline{Y}^{i}_{t})^{2}$
$\displaystyle\leqslant\widetilde{\mathbb{E}}^{i}\bigg{[}\int_{t}^{T}2K\overline{Y}^{i}\sum_{j\neq
i}^{\ell}\overline{Y}^{j}ds\;\bigg{|}\;\mathcal{F}^{W}_{t}\bigg{]}\leqslant
K\widetilde{\mathbb{E}}^{i}\bigg{[}\int_{t}^{T}\sum_{j=1}^{\ell}(\overline{Y}^{j})^{2}ds\;\bigg{|}\;\mathcal{F}^{W}_{t}\bigg{]}\leqslant
K\int_{t}^{T}\sum_{j=1}^{\ell}E_{s}^{j}ds,$

by the arithmetic-mean and geometric-mean inequality (AM-GM inequality), where

$E^{i}_{t}=\underset{\omega\in\Omega}{\operatorname*{ess\;sup}}\
{(\overline{Y}_{t}^{i})^{2}}.$

Taking essential supreme on both sides, we deduce

$E_{t}^{i}\leqslant K\int_{t}^{T}\sum_{j=1}^{\ell}E_{s}^{j}ds.$

Thus

$0\leqslant\sum_{j=1}^{\ell}E_{t}^{j}\leqslant
K\ell\int_{t}^{T}\sum_{j=1}^{\ell}E_{s}^{j}ds.$

We infer from Gronwall’s inequality that $\sum_{j=1}^{\ell}E_{t}^{j}=0$, so
$\overline{Y}_{t}^{i}=0$ a.s. for $t\in[0,T]$ and all $i\in\mathcal{M}$. This
completes the proof of the uniqueness. $\Box$

###### Remark 3.6

Please note that the solvability of (3.1) and (3.6) cannot be directly covered
by Fan, Hu and Tang [8] since the generators violate the locally Lipschitz
condition required in [8].

###### Theorem 3.7

Suppose $\gamma<0$ and Assumption 1 and 2 hold. Then there is a unique
uniformly positive solution $(P^{i},\Lambda^{i})_{i\in\mathcal{M}}$ to the
BSDE (3.1).

Proof: The proof is similar to the procedure of Theorem 3.5 with different
uniformly lower bounds and upper bounds. Hence we will only present how to
find these bounds. Other details are left to the interested readers.

Let $a^{\prime}>0$ be a large constant such that

$\displaystyle\frac{\gamma}{2(1-\gamma)}(b^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}b^{i}-\rho^{i}+\gamma
r^{i}\geqslant-a^{\prime}(1-\gamma),$ (3.9)

and

$\displaystyle-\rho^{i}+\gamma r^{i}-\gamma\varepsilon\leqslant a^{\prime},$
(3.10)

hold simultaneously for all $i\in\mathcal{M}$, where $\varepsilon>0$ is given
in Assumption 2. Denote $a_{1}^{\prime}:=e^{-a^{\prime}(1-\gamma)T}$ and
$a_{2}^{\prime}:=e^{a^{\prime}T}+\frac{\varepsilon^{\gamma}}{a^{\prime}}(e^{a^{\prime}T}-1)$.

Let $g:{\mathbb{R}}\rightarrow[0,1]$ be a smooth truncation function
satisfying $g(x)=0$ for $x\in(-\infty,\frac{1}{2}a^{\prime}_{1}]$, and
$g(x)=1$ for $x\in[a^{\prime}_{1},+\infty)$. Notice we have
$|xg(x)|\leqslant|x|$ for all $x\in{\mathbb{R}}$.

For integer $k\geqslant-\gamma\varepsilon$,
$(t,P,\Lambda)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{n}$,
$i\in\mathcal{M}$, define

$f^{k,i}(t,P,\Lambda)=\operatorname*{ess\;sup}_{\tilde{P}\in\mathbb{R},\tilde{\Lambda}\in\mathbb{R}^{n}}\Big{[}f^{i}(t,\tilde{P},\tilde{\Lambda})g(\tilde{P})-k|P-\tilde{P}|-k|\Lambda-\tilde{\Lambda}|\Big{]}.$

And consider the following $\ell$-dimensional BSDE:

$\displaystyle\begin{cases}dP^{k,i}=-\Big{[}f^{k,i}(P^{k,i},\Lambda^{k,i})-(\rho^{i}-\gamma
r^{i})P^{k,i}g(P^{k,i})+\sum_{j=1}^{\ell}q^{ij}P^{k,j}\Big{]}dt+(\Lambda^{k,i})^{\top}dW,\\\
P^{k,i}_{T}=1,\quad i\in\mathcal{M}.\end{cases}$

Since its generator is Lipschitz continuous, it admits a unique solution,
denoted by $\big{(}P^{k,i},\Lambda^{k,i}\big{)}_{i\in\mathcal{M}}$.

It is easy to verify that

$\displaystyle(\underline{P}^{i}_{t},\underline{\Lambda}^{i}_{t})=\bigg{(}\Big{(}e^{-a^{\prime}(T-t)}+\frac{1}{a^{\prime}}(1-e^{-a^{\prime}(T-t)})\Big{)}^{1-\gamma},0\bigg{)},\quad
i\in\mathcal{M}$ (3.11)

is a solution to the following BSDE system:

$\displaystyle\begin{cases}d\underline{P}^{i}=-\Big{[}-a^{\prime}(1-\gamma)\underline{P}^{i}g(\underline{P}^{i})+(1-\gamma)(\underline{P}^{i})^{-\frac{\gamma}{1-\gamma}}g(\underline{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}\Big{]}dt+(\underline{\Lambda}^{i})^{\prime}dW,\\\
\underline{P}^{i}_{T}=1,\quad i\in\mathcal{M}.\end{cases}$

Since $\gamma<0$, we have the following estimates for $f^{k,i}(t,P,\Lambda)$
with $P>0$:

$\displaystyle f^{k,i}(t,P,\Lambda)$ $\displaystyle\geqslant
f^{i}(t,P,\Lambda)g(P)$
$\displaystyle=\gamma\operatorname*{ess\;sup}_{(\pi,c)\in\Theta}\Big{[}-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{i}|^{2}P+\pi^{\prime}(Pb^{i}+\sigma^{i}\Lambda)+\frac{c^{\gamma}}{\gamma}-Pc\Big{]}g(P)$
$\displaystyle\geqslant\gamma\operatorname*{ess\;sup}_{(\pi,c)\in{\mathbb{R}}^{m}\times{\mathbb{R}}_{+}}\Big{[}-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{i}|^{2}P+\pi^{\prime}(Pb^{i}+\sigma^{i}\Lambda)+\frac{c^{\gamma}}{\gamma}-Pc\Big{]}g(P)$
$\displaystyle=\frac{\gamma}{2(1-\gamma)}\frac{1}{P}(Pb+\sigma\Lambda)^{\prime}(\sigma\sigma^{\prime})^{-1}(Pb+\sigma\Lambda)g(P)+(1-\gamma)P^{-\frac{\gamma}{1-\gamma}}g(P).$

Hence, we deduce

$\displaystyle\quad
f^{k,i}(t,\underline{P}^{i},\underline{\Lambda}^{i})-(\rho^{i}-\gamma
r^{i})\underline{P}^{i}g(\underline{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}$
$\displaystyle\geqslant\frac{\gamma}{2(1-\gamma)}(b^{i})^{\prime}(\sigma\sigma^{\prime})^{-1}b^{i}\underline{P}^{i}g(\underline{P}^{i})+(1-\gamma)\underline{P}^{-\frac{\gamma}{1-\gamma}}g(\underline{P}^{i})-(\rho^{i}-\gamma
r^{i})\underline{P}^{i}g(\underline{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}$
$\displaystyle\geqslant-a^{\prime}(1-\gamma)\underline{P}^{i}g(\underline{P}^{i})+(1-\gamma)\underline{P}^{-\frac{\gamma}{1-\gamma}}g(\underline{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}.$

By Lemma 3.3, we have

$a_{1}^{\prime}=e^{-a^{\prime}(1-\gamma)T}\leqslant\underline{P}^{i}_{t}\leqslant
P^{k,i}_{t},\quad i\in\mathcal{M}.$

To give an upper bound, we notice that

$\displaystyle(\bar{P}^{i},\bar{\Lambda}^{i})=\Big{(}e^{a^{\prime}(T-t)}+\frac{\varepsilon^{\gamma}}{a^{\prime}}(e^{a^{\prime}(T-t)}-1),0\Big{)},\quad
i\in\mathcal{M}$ (3.12)

is the unique solution to the following BSDE system:

$\displaystyle\begin{cases}d\bar{P}^{i}=-\Big{[}a^{\prime}\bar{P}^{i}+\varepsilon^{\gamma}+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}\Big{]}dt+(\bar{\Lambda}^{i})^{\prime}dW,\\\
\bar{P}^{i}_{T}=1,\quad i\in\mathcal{M}.\end{cases}$

Also, $(0\mathbf{1}_{m},\varepsilon)\in\Theta$ under Assumption 2, therefore
for $k\geqslant-\gamma\varepsilon$, we have

$\displaystyle f^{k,i}(t,P,\Lambda)$
$\displaystyle\leqslant\sup_{\tilde{P}\in\mathbb{R},\tilde{\Lambda}\in\mathbb{R}^{n}}\Big{[}(\varepsilon^{\gamma}-\gamma\tilde{P}\varepsilon)g(\tilde{P})-k|P-\tilde{P}|-k|\Lambda-\tilde{\Lambda}|\Big{]}$
$\displaystyle\leqslant\sup_{\tilde{P}\in\mathbb{R}}\Big{[}(\varepsilon^{\gamma}-\gamma\tilde{P}\varepsilon)g(\tilde{P})+\gamma\varepsilon|P-\tilde{P}|\Big{]}$
$\displaystyle\leqslant\sup_{\tilde{P}\in\mathbb{R}}\Big{[}(\varepsilon^{\gamma}-\gamma\tilde{P}\varepsilon)g(\tilde{P})+\gamma\varepsilon(|\tilde{P}|-|P|)\Big{]}$
$\displaystyle\leqslant\varepsilon^{\gamma}-\gamma\varepsilon|P|,$

where the third and fourth inequalities are due to that $\gamma<0$ and
$|xg(x)|\leqslant|x|$ for all $x\in{\mathbb{R}}$. Hence, using (3.10) and
$\bar{P}^{i}>0$, we get

$\displaystyle f^{k,i}(t,\bar{P}^{i},\bar{\Lambda}^{i})-(\rho^{i}-\gamma
r^{i})\bar{P}^{i}g(\bar{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}$
$\displaystyle\leqslant\varepsilon^{\gamma}-\gamma\varepsilon|\bar{P}^{i}|-(\rho^{i}-\gamma
r^{i})\bar{P}^{i}g(\bar{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}$
$\displaystyle\leqslant
a^{\prime}\bar{P}^{i}+\varepsilon^{\gamma}+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}.$

By Lemma 3.3 again, we obtain the upper bound

$P^{k,i}_{t}\leqslant\bar{P}^{i}_{t}\leqslant
e^{a^{\prime}T}+\frac{\varepsilon^{\gamma}}{a^{\prime}}(e^{a^{\prime}T}-1)=a_{2}^{\prime},\
t\in[0,T],\quad i\in\mathcal{M}.$

$\Box$

Based on the above two theorems, we can provide the complete answer to problem
(2.2).

###### Theorem 3.8

Suppose $\gamma\in(-\infty,0)\cup(0,1)$ and Assumptions 1 and 2 hold. Let
$(P^{i},\Lambda^{i})_{i\in\mathcal{M}}$ be the unique solution to (3.1). Then
the value function of the optimization problem (2.2) is given by

$V(x,i_{0})=\frac{1}{\gamma}x^{\gamma}P^{i_{0}}_{0},$

and $(\hat{\pi},\hat{c})\in\mathcal{U}$ is an optimal investment-consumption
pair if and only if

$(\hat{\pi}_{t},\hat{c}_{t})\in\operatorname*{argmax}_{(\pi,c)\in\Theta}\Big{[}-\frac{1-\gamma}{2}|\pi^{\prime}\sigma_{t}^{\alpha_{t}}|^{2}P_{t}^{\alpha_{t}}+\pi^{\prime}(P_{t}^{\alpha_{t}}b_{t}^{\alpha_{t}}+\sigma_{t}^{\alpha_{t}}\Lambda_{t}^{\alpha_{t}})+\frac{c^{\gamma}}{\gamma}-P_{t}^{\alpha_{t}}c\Big{]}.$

###### Remark 3.9

Let
$G:\Omega\times[0,T]\times\mathbb{R}^{m}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$
be a
$\mathcal{P}\otimes\mathcal{B}(\mathbb{R}^{m})\otimes\mathcal{B}(\mathbb{R}_{+})$-measurable
map, where $\mathcal{P}$ is the $\sigma$-field of predictable sets of
$\Omega\times[0,T]$, and $\mathcal{B}(\mathbb{R}^{m})$
($\mathcal{B}(\mathbb{R}_{+})$) is the Borelian $\sigma$-algebra on
$\mathbb{R}^{m}$ ($\mathbb{R}_{+}$). By

$\operatorname*{argmax}_{(\pi,c)\in\Theta}\Big{[}G(\omega,t,\pi,c)\Big{]},$

we denote the set of all predictable processes valued in $\Theta$ which attain
the essential supremum of $G$ w.r.t $(\pi,c)\in\Theta$. From a measurable
selection theorem (see e.g. [11, Lemma 11], [1, Corollary 18.14] or [7,
Proposition 2.4]), the set is not empty. If $\Theta$ is further convex, then
$\operatorname*{argmax}_{(\pi,c)\in\Theta}\Big{[}G(t,\omega,\pi,c)\Big{]}$
degenerates to a singleton set.

###### Remark 3.10

If $\Theta=\mathbb{R}^{n}\times\mathbb{R}_{+}$, then

$\hat{\pi}=\frac{1}{1-\gamma}(\sigma\sigma^{\prime})^{-1}(b+\sigma\frac{\Lambda}{P}),\quad\hat{c}=P^{\frac{1}{\gamma-1}}.$

Proof: For any $(\pi,c)\in\mathcal{U}$, applying Itô’s formula to
$\frac{1}{\gamma}e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}X_{t}^{\gamma}P_{t}^{\alpha_{t}}$,
(here we use Itô’s formula for Markovian chain; please refer to [12, Lemma
4.3]), we have

$\displaystyle\quad\frac{1}{\gamma}e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}X_{t}^{\gamma}P_{t}^{\alpha_{t}}+\int_{0}^{t}\frac{1}{\gamma}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}X_{s}^{\gamma}c_{s}^{\gamma}ds$
$\displaystyle=\frac{1}{\gamma}x^{\gamma}P_{0}^{i_{0}}+\int_{0}^{t}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}X_{s}^{\gamma}\Big{[}\frac{1}{\gamma}c^{\gamma}-cP-\frac{1-\gamma}{2}P|\pi^{\prime}\sigma|^{2}+\pi^{\prime}(Pb+\sigma\Lambda)-\frac{f}{\gamma}\Big{]}ds$
$\displaystyle\quad\;+\int_{0}^{t}\frac{1}{\gamma}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}X_{s}^{\gamma}(\Lambda^{\prime}+\gamma
P\pi^{\prime}\sigma)dW$
$\displaystyle\quad\;+\int_{0}^{t}\frac{1}{\gamma}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}X_{s}^{\gamma}\sum_{j,j^{\prime}\in\mathcal{M}}(P^{j}-P^{j^{\prime}})I_{\\{\alpha_{s-}=j^{\prime}\\}}d\tilde{N}^{j^{\prime}j}_{s},$

where $(N^{j^{\prime}j})_{j^{\prime}j\in\mathcal{M}}$ are independent Poisson
processes each with intensity $q^{j^{\prime}j}$, and
$\tilde{N}_{t}^{j^{\prime}j}=N_{t}^{j^{\prime}j}-q^{j^{\prime}j}t,\ t\geqslant
0$ are the corresponding compensated Poisson martingales under the filtration
$\mathcal{F}$. And we drop the superscript $\alpha_{s}$ in the above integral.

Noting that for any $(\pi,c)\in\mathcal{U}$, the wealth process $X$ is
continuous and strictly positive, hence bounded on $[0,T]$ away from $0$, a.s.
Therefore, the stochastic integrals in the last equation are local
martingales. Hence, there exists an increasing sequence of stopping times
$(\tau_{n})_{n\in\mathbb{N}}$ satisfying
$\lim_{n\rightarrow\infty}\tau_{n}=T$, a.s. such that

$\displaystyle\quad{\mathbb{E}}\Big{[}\frac{1}{\gamma}e^{-\int_{0}^{\tau_{n}}\rho_{s}^{\alpha_{s}}ds}X_{\tau_{n}}^{\gamma}P_{\tau_{n}}^{\alpha_{\tau_{n}}}+\int_{0}^{\tau_{n}}\frac{1}{\gamma}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}X_{s}^{\gamma}c_{s}^{\gamma}ds\Big{]}$
$\displaystyle=\frac{1}{\gamma}x^{\gamma}P_{0}^{i_{0}}+{\mathbb{E}}\int_{0}^{\tau_{n}}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}X_{s}^{\gamma}\Big{[}-\frac{1-\gamma}{2}P|\pi^{\prime}\sigma|^{2}+\pi^{\prime}(Pb+\sigma\Lambda)+\frac{1}{\gamma}c^{\gamma}-cP-\frac{f}{\gamma}\Big{]}ds$
$\displaystyle\leqslant\frac{1}{\gamma}x^{\gamma}P_{0}^{i_{0}},$

thanks to the definition of $f$.

Case $\gamma\in(0,1)$: In this case,
$\frac{1}{\gamma}e^{-\int_{0}^{\tau_{n}}\rho_{s}^{\alpha_{s}}ds}X_{\tau_{n}}^{\gamma}P_{\tau_{n}}^{\alpha_{\tau_{n}}}+\int_{0}^{\tau_{n}}\frac{1}{\gamma}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}X_{s}^{\gamma}c_{s}^{\gamma}ds$
is bounded from below by $0$. Passing to limit and applying Fatou’s lemma in
the above inequality yields

$\displaystyle{\mathbb{E}}\Big{[}\frac{1}{\gamma}e^{-\int_{0}^{T}\rho_{s}^{\alpha_{s}}ds}X_{T}^{\gamma}+\int_{0}^{T}\frac{1}{\gamma}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}X_{s}^{\gamma}c_{s}^{\gamma}ds\Big{]}$
$\displaystyle\leqslant\frac{1}{\gamma}x^{\gamma}P_{0}^{i_{0}},$ (3.13)

by virtue of the terminal condition $P_{T}^{\alpha_{T}}=1$. And the inequality
(3.13) becomes an equality for some $(\hat{\pi},\hat{c})$ if and only if

$(\hat{\pi},\hat{c})\in\operatorname*{argmax}_{(\pi,c)\in\Theta}\Big{[}-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{\alpha_{t}}|^{2}P+\pi^{\prime}(P^{\alpha_{t}}b^{\alpha_{t}}+\sigma^{\alpha_{t}}\Lambda^{\alpha_{t}})+\frac{c^{\gamma}}{\gamma}-P^{\alpha_{t}}c\Big{]}.$

It remains to show that $(\hat{\pi},\hat{c})\in\mathcal{U}$. Recall that the
unique solution of (3.1) $(P^{i},\Lambda^{i})\in
L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ for all
$i\in\mathcal{M}$. From (3.7), $|\hat{\pi}|\leqslant K(1+|Z^{\alpha_{t}}|)$,
hence $\hat{\pi}\in L^{2}_{\mathcal{F}}(0,T;\mathbb{R}^{m})$. Because $P$ is
uniformly positive and bounded, we have

$\displaystyle\quad-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{i}|^{2}P+\pi^{\prime}(Pb^{i}+\sigma^{i}\Lambda)+\frac{c^{\gamma}}{\gamma}-Pc$
$\displaystyle\leqslant\operatorname*{ess\;sup}_{\pi\in{\mathbb{R}}^{m}}\Big{[}-\frac{1-\gamma}{2}|\pi^{\prime}\sigma^{i}|^{2}P+\pi^{\prime}(Pb^{i}+\sigma^{i}\Lambda)\Big{]}+\frac{c^{\gamma}}{\gamma}-Pc$
$\displaystyle=\frac{1}{2}\frac{\gamma}{1-\gamma}\frac{1}{P}(Pb+\sigma\Lambda)^{\prime}(\sigma\sigma^{\prime})^{-1}(Pb+\sigma\Lambda)+\frac{c^{\gamma}}{\gamma}-Pc$
$\displaystyle\leqslant
K_{1}(1+|\Lambda|^{2})+\frac{c^{\gamma}}{\gamma}-K_{1}c$ $\displaystyle<0,$

if $c>K(1+|\Lambda|^{2})$ for sufficient large $K>0$. Hence $\hat{c}\leqslant
K(1+|\Lambda|^{2})$ so that $\int_{0}^{T}\hat{c}dt<\infty$, a.s. This implies
$(\hat{\pi},\hat{c})\in\mathcal{U}$.

Case $\gamma<0$: For any admissible $(\pi,c)\in\mathcal{U}$,
$(X^{\gamma})_{0\leqslant t\leqslant T}$ belongs to class (D) and
${\mathbb{E}}\big{[}\int_{0}^{T}(cX)^{\gamma}dt\big{]}<\infty$. So by
dominated convergence theorem, we also have (3.13). By a similar argument for
the previous case $\gamma\in(0,1)$, we can prove $\hat{\pi}\in
L^{2}_{\mathcal{F}}(0,T;\mathbb{R}^{m})$ and $\int_{0}^{T}\hat{c}dt<\infty$,
a.s. It remains to prove $(\hat{X}^{\gamma}_{t})_{0\leqslant t\leqslant T}\
\mbox{belongs to class (D) and
}{\mathbb{E}}\big{[}\int_{0}^{T}(\hat{c}\hat{X})^{\gamma}dt\big{]}<\infty$.

Since $\rho$ and $P$ are bounded, and

$\displaystyle\quad{\mathbb{E}}\bigg{[}e^{-\int_{0}^{\tau_{n}}\rho_{s}^{\alpha_{s}}ds}\hat{X}_{\tau_{n}}^{\gamma}P_{\tau_{n}}^{\alpha_{\tau_{n}}}+\int_{0}^{\tau_{n}}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}\hat{X}_{s}^{\gamma}\hat{c}_{s}^{\gamma}ds\bigg{]}=x^{\gamma}P_{0}^{i_{0}},$

Fatou’s Lemma yields

$\displaystyle{\mathbb{E}}\bigg{[}\hat{X}_{T}^{\gamma}+\int_{0}^{T}(\hat{c}\hat{X})^{\gamma}dt\bigg{]}<\infty.$
(3.14)

Denote
$\theta^{\alpha_{t}}:=(\sigma^{\alpha_{t}})^{\prime}(\sigma^{\alpha_{t}}(\sigma^{\alpha_{t}})^{\prime})^{-1}$
for all $i\in\mathcal{M}$. By Girsanov’s theorem,

$W^{\mathbb{Q}}_{t}:=W_{t}+\int_{0}^{t}\theta^{\alpha_{s}}_{s}ds$

is a Brownian motion under the measure $\mathbb{Q}$ defined by

$\frac{d\mathbb{Q}}{d\mathbb{P}}=\mathcal{E}\Big{(}-\int_{0}^{T}\theta^{\alpha_{s}}_{s}dW_{s}\Big{)}.$

Under the strategy $(\hat{\pi},\hat{c})$, we obtain from (2.1) that

$\displaystyle\hat{X}_{t}=xe^{\int_{0}^{t}(r-\hat{c})ds}\mathcal{E}\Big{(}\int_{0}^{t}\hat{\pi}^{\prime}\sigma
dW^{\mathbb{Q}}\Big{)}.$

Denote

$J_{t}:=e^{-\int_{0}^{t}\hat{c}ds}\mathcal{E}\Big{(}\int_{0}^{t}\hat{\pi}^{\prime}\sigma
dW^{\mathbb{Q}}\Big{)}.$

Thus it is sufficient to prove $(J^{\gamma}_{t})_{0\leqslant t\leqslant T}$
belongs to class (D), as $r$ is bounded.

It can be seen from (3.14) that ${\mathbb{E}}[J_{T}^{\gamma}]<\infty$. One
obtains for every stopping time $\tau\leqslant T$,

$\displaystyle J_{\tau}^{\frac{\gamma}{2}}$
$\displaystyle=\Big{\\{}{\mathbb{E}}^{\mathbb{Q}}\Big{[}e^{-\int_{0}^{\tau}\hat{c}ds}\mathcal{E}\Big{(}\int_{0}^{T}\hat{\pi}^{\prime}\sigma
dW^{\mathbb{Q}}\Big{)}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}\Big{\\}}^{\frac{\gamma}{2}}$
$\displaystyle\leqslant\Big{\\{}{\mathbb{E}}^{\mathbb{Q}}\Big{[}e^{-\int_{0}^{T}\hat{c}ds}\mathcal{E}\Big{(}\int_{0}^{T}\hat{\pi}^{\prime}\sigma
dW^{\mathbb{Q}}\Big{)}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}\Big{\\}}^{\frac{\gamma}{2}}=\Big{\\{}{\mathbb{E}}^{\mathbb{Q}}\Big{[}J_{T}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}\Big{\\}}^{\frac{\gamma}{2}}\leqslant{\mathbb{E}}^{\mathbb{Q}}\Big{[}J_{T}^{\frac{\gamma}{2}}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]},$
(3.15)

where we used that $\hat{c}>0$ and $\gamma<0$ in the first inequality, and
Jensen’s inequality in the second one. By the Cauchy-Schwarz inequality and
the boundedness of $\theta$,

$\displaystyle{\mathbb{E}}^{\mathbb{Q}}\Big{[}J_{T}^{\frac{\gamma}{2}}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}$
$\displaystyle={\mathbb{E}}\Big{[}J_{T}^{\frac{\gamma}{2}}\mathcal{E}\Big{(}-\int_{\tau}^{T}\theta_{s}^{\alpha_{s}}dW_{s}\Big{)}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}$
$\displaystyle\leqslant\Big{\\{}{\mathbb{E}}\Big{[}J_{T}^{\gamma}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}\Big{\\}}^{\frac{1}{2}}\Big{\\{}{\mathbb{E}}\Big{[}\mathcal{E}\Big{(}-\int_{\tau}^{T}\theta_{s}^{\alpha_{s}}dW_{s}\Big{)}^{2}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}\Big{\\}}^{\frac{1}{2}}\leqslant
K\Big{\\{}{\mathbb{E}}\Big{[}J_{T}^{\gamma}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}\Big{\\}}^{\frac{1}{2}}.$
(3.16)

Then (3) and (3) imply that $(J^{\gamma}_{t})_{0\leqslant t\leqslant T}$
belongs to class (D). $\Box$

##  4 Logarithmic utility

Let us now turn to the logarithmic utility $U(x)=\ln x,\ x>0$. The admissible
set is now defined as

$\displaystyle\mathcal{U}$ $\displaystyle=\Big{\\{}(\pi,c)\;\Big{|}\;\pi\in
L^{2}_{\mathcal{F}}(0,T;\mathbb{R}^{m}),\ c\in
L^{1}_{\mathcal{F}}(0,T;\mathbb{R}_{+}),\
(\pi_{t}(\omega),c_{t}(\omega))\in\Theta,\ a.e.,\ a.s,$
$\displaystyle\qquad\qquad\quad(\ln X_{t})_{0\leqslant t\leqslant T}\
\mbox{and}\ \Big{(}\int_{0}^{t}\ln(c_{s}X_{s})ds\Big{)}_{0\leqslant t\leqslant
T}\ \mbox{belong to class (D)}\Big{\\}}.$

Clearly $\mathcal{U}$ would be empty if $c$ could not take positive value.
Hence, it is reasonable to put the following assumption.

###### Assumption 3

There exists $\varepsilon>0$ such that
$(0\mathbf{1}_{m},\varepsilon)\in\Theta$.

To solve problem (2.2), we introduce the following BSDE system:

$\displaystyle\begin{cases}dh_{t}^{i}=-(1-\rho^{i}h_{t}^{i}+\sum\limits_{j=1}^{\ell}q^{ij}h^{j})dt+(\eta^{i})^{\prime}dW,\\\
h_{T}^{i}=1,\quad i\in\mathcal{M}.\end{cases}$ (4.1)

This is a linear BSDE system with bounded coefficients, so it has a unique
solution $(h^{i},\eta^{i})_{i\in\mathcal{M}}$ such that $(h^{i},\eta^{i})\in
L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$.

We claim that $h^{i}$ is uniformly positive. Indeed, under Assumption 1, there
exists a constant $k>0$ such that $-\rho^{i}\geqslant-k$ for all
$i\in\mathcal{M}$. Notice that

$\Big{(}e^{-k(T-t)}+\frac{1-e^{-k(T-t)}}{k},0\Big{)}$

is a solution to the following BSDE:

$\displaystyle\begin{cases}dh_{t}=-(1-kh_{t})dt+\eta^{\prime}_{t}dW,\\\
h_{T}=1,\end{cases}$

From Lemma 3.3, we have $h^{i}_{t}\geqslant
e^{-k(T-t)}+\frac{1-e^{-k(T-t)}}{k}\geqslant e^{-kT}$.

For the unique solution $(h^{i},\eta^{i})_{i\in\mathcal{M}}$ of (4.1),
consider the following BSDE system:

$\displaystyle\begin{cases}dP^{i}=-\Big{[}f^{i}-\rho^{i}P^{i}+r^{i}h^{i}+\sum\limits_{j=1}^{\ell}q^{ij}P^{j}\Big{]}dt+(\Lambda^{i})^{\prime}dW\\\
P^{i}_{T}=0,\quad i\in\mathcal{M},\end{cases}$ (4.2)

where

$f^{i}:=\operatorname*{ess\;sup}_{(\pi,c)\in\Theta}\Big{[}-\frac{1}{2}h^{i}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(h^{i}b^{i}+\sigma^{i}\eta^{i})+\ln
c-h^{i}c\Big{]}.$

As $(0\mathbf{1}_{m},\varepsilon)\in\Theta$, and $h^{i}\in
L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})$, we have

$\displaystyle f^{i}\geqslant\ln\varepsilon-h^{i}\varepsilon.$ (4.3)

On the other hand,

$\displaystyle f^{i}$
$\displaystyle\leqslant\operatorname*{ess\;sup}\limits_{(\pi,c)\in\mathbb{R}^{m}\times\mathbb{R}_{+}}\Big{[}-\frac{1}{2}h^{i}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(h^{i}b^{i}+\sigma^{i}\eta^{i})+\ln
c-h^{i}c\Big{]}$
$\displaystyle=\frac{1}{2h^{i}}(h^{i}b^{i}+\sigma^{i}\eta^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}(h^{i}b^{i}+\sigma^{i}\eta^{i})-1-\ln
h^{i}.$

Thus

$\displaystyle|f^{i}|\leqslant K(1+|\eta^{i}|^{2}).$

Whence (4.2) is an $\ell$-dimensional linear BSDE system with unbounded
coefficients. By [15, Theorem 3.6], the system of (4.2) admits a unique
solution $(P^{i},\Lambda^{i})_{i\in\mathcal{M}}$ such that

$(P^{i},\Lambda^{i})\in L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n}),\ \mbox{for
all}\quad i\in\mathcal{M}.$

###### Remark 4.1

Set
$(Y^{i},Z^{i})=(-\frac{P^{i}}{h^{i}},\frac{P^{i}}{(h^{i})^{2}}\eta^{i}-\frac{1}{h^{i}}\Lambda^{i})$,
then $(Y^{i},Z^{i})$ satisfies

$\displaystyle\begin{cases}dY^{i}=-\Big{\\{}\operatorname*{ess\;inf}\limits_{(\pi,c)\in\Theta}\Big{[}\frac{1}{2}|\pi^{\prime}\sigma^{i}|^{2}-\pi^{\prime}(b^{i}+\frac{1}{h^{i}}\sigma^{i}\eta^{i})-\frac{\ln
c}{h^{i}}+c\Big{]}-\frac{1}{h^{i}}(1+\sum\limits_{j=1}^{\ell}q^{ij}h^{j})Y^{i}\\\
\qquad\qquad\;+\frac{1}{h^{i}}({\color[rgb]{1,0,0}\eta^{i}})^{\prime}Z^{i}-r^{i}+\frac{1}{h^{i}}\sum\limits_{j=1}^{\ell}q^{ij}h^{j}Y^{j}\Big{\\}}dt+(Z^{i})^{\prime}dW,\\\
Y^{i}_{T}=0,\quad i\in\mathcal{M}.\end{cases}$ (4.4)

If there is no regime switching and the $\rho$ is a constant, then
$\eta^{i}\equiv 0$ and (4.4) degenerates to (4.6) in [4].

The following lemma can be found in Page 26 of [20].

###### Lemma 4.2

Suppose $\phi\in L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$.
Then

$\displaystyle{\mathbb{E}}\left[\Big{(}\int_{0}^{T}|\phi_{s}|^{2}ds\Big{)}^{j}\right]<\infty,\quad\mbox{for
any $j>0$}.$

###### Theorem 4.3

Suppose Assumptions 1 and 3 hold. Let $(h^{i},\eta^{i})_{i\in\mathcal{M}}$ and
$(Y^{i},Z^{i})_{i\in\mathcal{M}}$ be the unique solutions to (4.1) and (4.4),
respectively. Then the value function of the optimization problem (2.2) is
given by

$V(x,i_{0})=h_{0}^{i_{0}}\ln x+P_{0}^{i_{0}},$

and $(\hat{\pi},\hat{c})\in\mathcal{U}$ is an optimal consumption-investment
pair if and only if

$(\hat{\pi}_{t},\hat{c}_{t})\in\operatorname*{argmax}_{(\pi,c)\in\Theta}\Big{[}-\frac{1}{2}h_{t}^{\alpha_{t}}|\pi^{\prime}\sigma_{t}^{\alpha_{t}}|^{2}+\pi^{\prime}(h_{t}^{\alpha_{t}}b_{t}^{\alpha_{t}}+\sigma_{t}^{\alpha_{t}}\eta_{t}^{\alpha_{t}})+\ln
c-h_{t}^{\alpha_{t}}c\Big{]}.$

###### Remark 4.4

If $\Theta=\mathbb{R}^{m}\times\mathbb{R}_{+}$, then the optimal investment-
consumption pair is uniquely given by

$\hat{\pi}_{t}=(\sigma^{\alpha_{t}}(\sigma^{\alpha_{t}})^{\prime})^{-1}(b^{\alpha_{t}}+\frac{1}{h^{\alpha_{t}}}\sigma^{\alpha_{t}}\eta^{\alpha_{t}}),\quad\hat{c}_{t}=\frac{1}{h_{t}^{\alpha_{t}}}.$

Proof: For any $(\pi,c)\in\mathcal{U}$, applying Itô’s formula to
$e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}(h_{t}^{\alpha_{t}}\ln
X_{t}+P_{t}^{\alpha_{t}})$, we have

$\displaystyle\quad
e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}(h_{t}^{\alpha_{t}}\ln
X_{t}+P_{t}^{\alpha_{t}})+\int_{0}^{t}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}\ln(c_{s}X_{s})ds$
$\displaystyle=h_{0}^{i_{0}}\ln
x+P_{0}^{i_{0}}+\int_{0}^{t}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}\Big{[}\ln
c-ch-\frac{1}{2}|\pi^{\prime}\sigma|^{2}+\pi^{\prime}(h\eta+\sigma\eta)-f^{\alpha_{s}}\Big{]}ds$
$\displaystyle\qquad+\int_{0}^{t}e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}\Big{[}h\sigma^{\prime}\pi+\eta\ln
X+\Lambda\Big{]}^{\prime}dW$
$\displaystyle\qquad+\int_{0}^{t}e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}\sum_{j,j^{\prime}\in\mathcal{M}}[(h^{j}\ln
X+P^{j})-(h^{j^{\prime}}\ln
X+P^{j^{\prime}})]I_{\\{\alpha_{s-}=j^{\prime}\\}}d\tilde{N}^{j^{\prime}j}_{s}.$

Then by the definition of $f^{i}$, the process

$e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}(h_{t}^{\alpha_{t}}\ln
X_{t}+P_{t}^{\alpha_{t}})+\int_{0}^{t}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}\ln(c_{s}X_{s})ds$

is a local supermartingale. Hence, there exists a sequence of increasing
stopping times $(\tau_{n})_{n\in\mathbb{N}}$ satisfying
$\lim_{n\rightarrow\infty}\tau_{n}=T$, a.s. such that

$\displaystyle{\mathbb{E}}\Big{[}e^{-\int_{0}^{\tau_{n}}\rho_{s}^{\alpha_{s}}ds}(h_{\tau_{n}}^{\alpha_{\tau_{n}}}\ln
X_{\tau_{n}}+P_{\tau_{n}}^{\alpha_{\tau_{n}}})+\int_{0}^{\tau_{n}}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}\ln(c_{s}X_{s})ds\Big{]}\leqslant
h_{0}^{i_{0}}\ln x+P_{0}^{i_{0}}.$

Recalling the admissibility of $(\pi,c)$, we deduce from the dominated
convergence theorem that

$\displaystyle{\mathbb{E}}\Big{[}e^{-\int_{0}^{T}\rho_{s}^{\alpha_{s}}ds}(h_{T}^{\alpha_{T}}\ln
X_{T}+P_{T}^{\alpha_{T}})+\int_{0}^{T}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}\ln(c_{s}X_{s})ds\Big{]}\leqslant
h_{0}^{i_{0}}\ln x+P_{0}^{i_{0}},$ (4.5)

with the equality holds for some $(\hat{\pi},\hat{c})$ if and only if

$(\hat{\pi}_{t},\hat{c}_{t})\in\operatorname*{argmax}_{(\pi,c)\in\Theta}\Big{[}-\frac{1}{2}h^{i}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(h^{i}b^{i}+\sigma^{i}\eta^{i})+\ln
c-h^{i}c\Big{]}.$

It remains to prove $(\hat{\pi},\hat{c})\in\mathcal{U}$ for such a pair. 444
Compared with [4], a new, non-zero term $\eta\in
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ emerges (In [4],
$\rho$ is a constant, hence $\eta=0$). So we need conditions like class (D) to
apply the dominated convergence theorem in (4.5). But this brings difficulty
in proving the admissibility of $(\hat{\pi},\hat{c})$.

From (4.3) and $h^{i}\in L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})$,
$f^{i}\geqslant-K_{1}$ for some $K_{1}>0$. On the other hand, Assumption 1 and
$h^{i}\geqslant e^{-kT}$ imply

$\displaystyle\quad-\frac{1}{2}h^{i}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(h^{i}b^{i}+\sigma^{i}\eta^{i})+\ln
c-h^{i}c$
$\displaystyle\leqslant-\frac{1}{2}h^{i}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(h^{i}b^{i}+\sigma^{i}\eta^{i})+\operatorname*{ess\;sup}_{c\in{\mathbb{R}}_{+}}(\ln
c-h^{i}c)$
$\displaystyle=-\frac{1}{2}h^{i}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(h^{i}b^{i}+\sigma^{i}\eta^{i})-\ln
h^{i}-1$ $\displaystyle\leqslant-
K_{2}|\pi|^{2}+K_{3}|\pi|(1+|\eta^{i}|)+K_{4},$

for some positive constants $K_{2},\ K_{3},\ K_{4}$. Thus

$\displaystyle\quad-\frac{1}{2}h^{i}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(h^{i}b^{i}+\sigma^{i}\eta^{i})+\ln
c-h^{i}c<-K_{1}\leqslant f^{i},$

if $|\pi|>K(1+|\eta^{i}|)$ with $K$ being sufficient large, which implies

$\displaystyle|\hat{\pi}|\leqslant K(1+|\eta^{i}|).$ (4.6)

Similarly, since $h^{i}$ is uniformly positive, we have for some
$K_{5},K_{6}>0$,

$\displaystyle\quad-\frac{1}{2}h^{i}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(h^{i}b^{i}+\sigma^{i}\eta^{i})+\ln
c-h^{i}c$
$\displaystyle\leqslant\operatorname*{ess\;sup}_{\pi\in{\mathbb{R}}^{m}}\Big{[}-\frac{1}{2}h^{i}|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(h^{i}b^{i}+\sigma^{i}\eta^{i})\Big{]}+\ln
c-K_{5}c$
$\displaystyle=\frac{1}{2h^{i}}(h^{i}b^{i}+\sigma^{i}\eta^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}(h^{i}b^{i}+\sigma^{i}\eta^{i})+\ln
c-K_{5}c$ $\displaystyle\leqslant K_{6}(1+|\eta^{i}|^{2})+\ln c-K_{5}c$
$\displaystyle<-K_{1}$ $\displaystyle\leqslant f^{i},$

if $c\geqslant K(1+|\eta^{i}|^{2})$ or $\ln c\leqslant-K(1+|\eta^{i}|^{2})$
for sufficient large $K$. Therefore,

$\displaystyle e^{-K(1+|\eta^{i}|^{2})}\leqslant\hat{c}\leqslant
K(1+|\eta^{i}|^{2}),$ (4.7)

and consequently,

$\displaystyle|\ln\hat{c}|\leqslant K(1+|\eta^{i}|^{2}).$ (4.8)

From (2.1),

$\displaystyle\ln\hat{X}_{t}$
$\displaystyle=\ln(x)+\int_{0}^{t}(r+\hat{\pi}^{\prime}b-\hat{c}-\frac{1}{2}|\hat{\pi}^{\prime}\sigma|^{2})ds+\int_{0}^{t}\hat{\pi}^{\prime}\sigma
dW.$

It follows from the boundedness of coefficients and the AM-GM inequality that

$\displaystyle\sup_{t\leqslant T}|\ln\hat{X}_{t}|$ $\displaystyle\leqslant
K+K\int_{0}^{T}(|\hat{c}|+|\hat{\pi}|^{2})ds+K\sup_{t\leqslant
T}\bigg{|}\int_{0}^{t}\hat{\pi}^{\prime}\sigma dW\Big{|}.$

Using the AM-GM inequality, the Burkholder-Davis-Gundy inequality, (4.6),
(4.7), and Lemma 4.2, we have

$\displaystyle{\mathbb{E}}\Big{[}\sup_{\tau\leqslant
T}(\ln\hat{X}_{\tau})^{2}\Big{]}$ $\displaystyle\leqslant
K+K{\mathbb{E}}\bigg{[}\Big{(}\int_{0}^{T}(|\hat{c}|+|\hat{\pi}|^{2})ds\Big{)}^{2}+\sup_{t\leqslant
T}\bigg{|}\int_{0}^{t}\hat{\pi}^{\prime}\sigma dW\bigg{|}^{2}\bigg{]}$
$\displaystyle\leqslant
K+K{\mathbb{E}}\bigg{[}\Big{(}\int_{0}^{T}(1+|\eta^{i}|^{2})ds\Big{)}^{2}+\int_{0}^{T}|\hat{\pi}^{\prime}\sigma|^{2}ds\bigg{]}$
$\displaystyle\leqslant
K+K{\mathbb{E}}\bigg{[}\Big{(}\int_{0}^{T}(1+|\eta^{i}|^{2})ds\Big{)}^{2}+\int_{0}^{T}(1+|\eta^{i}|^{2})ds\bigg{]}$
$\displaystyle\leqslant K.$

This shows that $(\ln X_{t})_{0\leqslant t\leqslant T}$ belongs to class (D).
By the same argument and using (4.8), we can prove that
$\Big{(}\int_{0}^{t}\ln(c_{s}X_{s})ds\Big{)}_{0\leqslant t\leqslant T}$
belongs to class (D) as well. $\Box$

##  5 Exponential utility

In this case, we do not consider proportional consumption-investment
strategies. Instead, let $\pi_{j,t}$ be the _amount_ invest in the $j$th risky
asset, $j=1,\ldots,m$, and let $c_{t}$ be the _amount_ to consume. Then the
investor’s self-financing wealth process $X$ corresponding to a consumption-
investment strategy $(\pi,c)$ is the unique strong solution of the SDE:

$\displaystyle\begin{cases}dX_{t}=(r_{t}X_{t}+\pi_{t}^{\prime}b_{t}^{\alpha_{t}}-c_{t})dt+\pi_{t}^{\prime}\sigma_{t}^{\alpha_{t}}dW_{t},\\\
X_{0}=x,\ \alpha_{0}=i_{0}.\end{cases}$ (5.1)

Let $\Pi$ be a given closed nonempty set in $\mathbb{R}^{m}$ with $0\in\Pi$ to
represent the constraint set for portfolios. We do not put constraint on
consumptions, so $-c_{t}$ can be interpreted as income if $c_{t}<0$. The
admissible consumption-investment set is defined as

$\displaystyle\mathcal{U}$
$\displaystyle=\bigg{\\{}(\pi,c)\;\bigg{|}\;\int_{0}^{T}(|\pi_{t}|^{2}+|c_{t}|)dt<\infty,\
(\pi_{t}(\omega),c_{t}(\omega))\in\Pi\times\mathbb{R},\ a.e.,\ a.s,$
$\displaystyle\qquad\qquad\quad\;\mbox{and}\ (e^{-\beta
h_{t}X_{t}^{\pi,c}})_{0\leqslant t\leqslant T}\ \mbox{ belongs to class
(D)}\bigg{\\}},$

where the process $h_{t}$ will be specified case by case in the following
subsections.

The investor’s problem is to maximize

$\displaystyle J(x,i_{0};\pi,c):$
$\displaystyle={\mathbb{E}}\bigg{[}\int_{0}^{T}e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}U(c_{t})dt+e^{-\int_{0}^{T}\rho_{s}^{\alpha_{s}}ds}U(X_{T})\bigg{]},\quad\mathrm{s.t.}\;\;(\pi,c)\in\mathcal{U},$
(5.2)

and determine the value function

$V(x,i_{0}):=\sup_{(\pi,c)\in\mathcal{U}}J(x,i_{0};\pi,c),$

where the utility is exponential and given by

$U(x)=-\exp(-\beta x),\quad x\in\mathbb{R},\quad\beta>0.$

It turns out that whether the interest rate is deterministic or random is
crucial, we have to deal with these two cases separately.

###  5.1 Case I: Deterministic interest rate

###### Assumption 4

The process $r$ is a deterministic bounded measurable function of $t$. For all
$i\in\mathcal{M}$,

$\displaystyle\mu^{i}\in
L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}^{m}),\quad\sigma^{i}\in
L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}^{m\times n}),\quad\rho^{i}\in
L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}),\ $

and $\sigma^{i}(\sigma^{i})^{\prime}\geqslant\delta I_{m}$ with some constant
$\delta>0$.

Define

$\displaystyle
h_{t}=\Big{(}e^{-\int_{t}^{T}r_{s}ds}+\int_{t}^{T}e^{-\int_{t}^{s}r_{u}du}ds\Big{)}^{-1}>0.$
(5.3)

Then under Assumption 4, $h$ is the unique bounded uniformly positive solution
to the following ODE:

$\displaystyle\begin{cases}dh_{t}=-h_{t}(r_{t}-h_{t})dt,\\\
h_{T}=1.\end{cases}$

Consider the following BSDE system:

$\displaystyle\begin{cases}dP^{i}=-\Big{\\{}f^{i}(P^{i},\Lambda^{i})-hP^{i}\ln
P^{i}-\rho^{i}P^{i}+h(1-\ln
h)P^{i}+\sum_{j=1}^{\ell}q^{ij}P^{j}\Big{\\}}dt+(\Lambda^{i})^{\prime}dW,\\\
P^{i}_{T}=1,\quad i\in\mathcal{M},\end{cases}$ (5.4)

where

$\displaystyle f^{i}(P,\Lambda):=\beta
h\operatorname*{ess\;inf}_{\pi\in\Pi}\Big{[}\frac{1}{2}\beta
hP^{i}|\pi^{\prime}\sigma^{i}|^{2}-\pi^{\prime}(P^{i}b^{i}+\sigma^{i}\Lambda^{i})\Big{]}.$

Thanks to $0\in\Pi$, we have $f^{i}\leqslant 0$.

Solutions to (5.4) are defined as in Definition 3.2.

###### Theorem 5.1

Under Assumption 4, there is a unique uniformly positive solution
$(P^{i},\Lambda^{i})_{i\in\mathcal{M}}$ to the BSDE (5.4).

Proof: We will only present how to find the uniformly bounds. Details are left
to interested readers.

Let $a_{1}>0$, $a_{2}>0$, $a_{3}>0$ be three constants such that

$\displaystyle-\rho^{i}+h(1-\ln h)\leqslant a_{1},$
$\displaystyle\frac{1}{2}(b^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}b^{i}-\rho^{i}+h(1-\ln
h)\geqslant-a_{2},$ $\displaystyle h\geqslant a_{3}>0,\ \forall
i\in\mathcal{M}.$ (5.5)

Denote $a:=e^{a_{1}T}$ and
$\epsilon:=e^{-\frac{a_{2}}{a_{3}}(1-e^{-a_{3}T})}$. Note $\epsilon<1<a$.

Let $g:{\mathbb{R}}\to[0,1]$ be a smooth truncation function satisfying
$g(x)=0$ for $x\in(-\infty,\epsilon/2]\cup[2a,\infty)$, and $g(x)=1$ for
$x\in[\epsilon,a]$.

For $k\geqslant 1$,
$(t,P,\Lambda)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{n}$,
$i\in\mathcal{M}$, define

$f^{k,i}(t,P,\Lambda)=\sup_{\tilde{P}\in\mathbb{R},\tilde{\Lambda}\in\mathbb{R}^{n}}\Big{[}f^{i}(t,\tilde{P},\tilde{\Lambda})g(\tilde{P})-k|P-\tilde{P}|-k|\Lambda-\tilde{\Lambda}|\Big{]}.$

Let $\big{(}P^{k,i},\Lambda^{k,i}\big{)}_{i\in\mathcal{M}}$ denote the unique
solution to the following $\ell$-dimensional BSDE system with a Lipschitz
generator

$\displaystyle\begin{cases}dP^{k,i}=-\Big{[}f^{k,i}(P^{k,i},\Lambda^{k,i,+})-hP^{k,i}(\ln
P^{k,i})g(P^{k,i})-\rho^{i}P^{k,i}g(P^{k,i})\\\ \qquad\qquad\quad+h(1-\ln
h)P^{k,i}g(P^{k,i})+\sum_{j=1}^{\ell}q^{ij}P^{k,j}\Big{]}dt+(\Lambda^{k,i})^{\top}dW,\\\
P^{k,i}_{T}=1,\quad i\in\mathcal{M}.\end{cases}$

It is direct to verify that

$\displaystyle(\underline{P}^{i}_{t},\underline{\Lambda}^{i}_{t})=\Big{(}e^{-\frac{a_{2}}{a_{3}}(1-e^{-a_{3}(T-t)})},0\Big{)},\quad
i\in\mathcal{M}$ (5.6)

is the unique solution to the following BSDE system:

$\displaystyle\begin{cases}d\underline{P}^{i}=-\Big{[}-a_{2}\underline{P}^{i}g(\underline{P}^{i})-a_{3}\underline{P}^{i}(\ln\underline{P}^{i})g(\underline{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}\Big{]}dt+(\underline{\Lambda}^{i})^{\prime}dW,\\\
\underline{P}^{i}_{T}=1,\quad i\in\mathcal{M}.\end{cases}$

Notice that $\underline{P}^{i}\in(0,1)$, and

$\displaystyle f^{k,i}(t,P,\Lambda)$ $\displaystyle\geqslant
f^{i}(t,P,\Lambda)g(P)$ $\displaystyle=\beta
h\operatorname*{ess\;inf}_{\pi\in\Pi}\Big{[}\frac{1}{2}\beta
hP^{i}|\pi^{\prime}\sigma^{i}|^{2}-\pi^{\prime}(P^{i}b^{i}+\sigma^{i}\Lambda^{i})\Big{]}g(P)$
$\displaystyle\geqslant\beta
h\operatorname*{ess\;inf}_{\pi\in\mathbb{R}^{m}}\Big{[}\frac{1}{2}\beta
hP^{i}|\pi^{\prime}\sigma^{i}|^{2}-\pi^{\prime}(P^{i}b^{i}+\sigma^{i}\Lambda^{i})\Big{]}g(P)$
$\displaystyle=\frac{1}{2}(b^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}b^{i}Pg(P)+(b^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}\sigma\Lambda
g(P)$
$\displaystyle\qquad+\frac{1}{2P}\Lambda^{\prime}(\sigma^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}\sigma^{i}\Lambda
g(P),$

we deduce from (5.1) that

$\displaystyle\quad
f^{k,i}(\underline{P}^{i},\underline{\Lambda}^{i})-h\underline{P}^{i}(\ln\underline{P}^{i})g(\underline{P}^{i})-\rho^{i}\underline{P}^{i}g(\underline{P}^{i})+h(1-\ln
h)\underline{P}^{i}g(\underline{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{i}$
$\displaystyle\geqslant\frac{1}{2}(b^{i})^{\prime}(\sigma^{i}(\sigma^{i})^{\prime})^{-1}b^{i}\underline{P}^{i}g(\underline{P}^{i})-h\underline{P}^{i}(\ln\underline{P}^{i})g(\underline{P}^{i})-\rho^{i}\underline{P}^{i}g(\underline{P}^{i})+h(1-\ln
h)\underline{P}^{i}g(\underline{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{i}$
$\displaystyle\geqslant-
a_{2}\underline{P}^{i}g(\underline{P}^{i})-a_{3}\underline{P}^{i}(\ln\underline{P}^{i})g(\underline{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\underline{P}^{j}.$

By Lemma 3.3, we have

$\displaystyle P^{k,i}_{t}\geqslant\underline{P}^{i}_{t}\geqslant
e^{-\frac{a_{2}}{a_{3}}(1-e^{-a_{3}T})}=\epsilon,\quad i\in\mathcal{M}.$ (5.7)

This gives a uniform lower bound.

To give a uniform upper bound, we notice

$\displaystyle(\bar{P}^{i}_{t},\bar{\Lambda}^{i}_{t})=(e^{a_{1}(T-t)},0),\quad
i\in\mathcal{M},$ (5.8)

is a solution to the following BSDE system:

$\displaystyle\begin{cases}d\bar{P}^{i}=-\Big{[}a_{1}\bar{P}^{i}g(\bar{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}\Big{]}dt+(\bar{\Lambda}^{i})^{\prime}dW,\\\
\bar{P}^{i}_{T}=1,\quad i\in\mathcal{M}.\end{cases}$

Since $0\in\Pi$, we have $f^{i}(t,P,\Lambda)\leqslant 0$, and hence
$f^{k,i}(t,P,\Lambda)\leqslant 0$. Notice that $\bar{P}^{i}_{t}\geqslant 1$,
thus we have

$\displaystyle\quad
f^{k,i}(\bar{P}^{i},\bar{\Lambda}^{i})-h\bar{P}^{i}(\ln\bar{P}^{i})g(\bar{P}^{i})-\rho^{i}\bar{P}^{i}g(\bar{P}^{i})+h(1-\ln
h)\bar{P}^{i}g(\bar{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}$
$\displaystyle\leqslant
a_{1}\bar{P}^{i}g(\bar{P}^{i})+\sum_{j=1}^{\ell}q^{ij}\bar{P}^{j}.$

By Lemma 3.3, we have

$\displaystyle P^{k,i}_{t}\leqslant\bar{P}^{i}_{t}\leqslant e^{a_{1}T}=a,\quad
i\in\mathcal{M}.$ (5.9)

This gives a uniform upper bound. $\Box$

Set $Y^{i}=-\frac{1}{\beta}\ln P^{i}$, $Z^{i}=-\frac{\Lambda^{i}}{\beta
P^{i}}$, then $(Y^{i},Z^{i})$ is the unique solution to

$\displaystyle\begin{cases}dY^{i}=-\Big{[}F^{i}(Z^{i})-hY^{i}-\frac{\beta}{2}|Z^{i}|^{2}+\frac{\rho^{i}}{\beta}-\frac{h}{\beta}(1-\ln
h)\\\
\qquad\qquad-\frac{1}{\beta}\sum_{j=1}^{\ell}q^{ij}\Big{(}e^{-\beta(Y^{j}-Y^{i})}-1\Big{)}\Big{]}dt+(Z^{i})^{\prime}dW,\\\
Y^{i}_{T}=0,\quad i\in\mathcal{M},\end{cases}$ (5.10)

where

$F^{i}(z):=h\operatorname*{ess\;sup}_{\pi\in\Pi}\Big{[}-\frac{1}{2}\beta
h|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(b^{i}-\beta\sigma^{i}z)\Big{]},\quad
z\in{\mathbb{R}}^{n}.$

From above, we immediately have

###### Theorem 5.2

Under Assumption 4, there is a unique solution
$(Y^{i},Z^{i})_{i\in\mathcal{M}}$ to the BSDE (5.10) such that
$(Y^{i},Z^{i})\in L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ for all
$i\in\mathcal{M}$.

Now we can present the solution to problem (5.2).

###### Theorem 5.3

Suppose Assumption 4 holds. Let $h$ be defined in (5.3) and
$(Y^{i},Z^{i})_{i\in\mathcal{M}}$ be the unique solution to (5.10). Then the
value function of the optimization problem (5.2) is given by

$V(x,i_{0})=-e^{-\beta(h_{0}x+Y_{0}^{i_{0}})},$

and the optimal portfolio is

$\displaystyle\hat{\pi}_{t}\in\operatorname*{argmax}_{\pi\in\Pi}\Big{[}-\frac{1}{2}\beta
h_{t}|\pi^{\prime}\sigma_{t}^{\alpha_{t}}|^{2}+\pi^{\prime}(b_{t}^{\alpha_{t}}-\beta\sigma_{t}^{\alpha_{t}}Z_{t}^{\alpha_{t}})\Big{]},$

the optimal consumption is

$\displaystyle\hat{c}_{t}=h_{t}X_{t}+Y_{t}^{\alpha_{t}}-\frac{1}{\beta}\ln
h_{t}.$

Proof: For any $(\pi,c)\in\mathcal{U}$, applying Itô’s formula to
$-e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}e^{-\beta(h_{t}X_{t}+Y_{t}^{\alpha_{t}})}$,
we have

$\displaystyle\quad-e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}e^{-\beta(h_{t}X_{t}+Y_{t}^{\alpha_{t}})}-\int_{0}^{t}e^{-\int_{0}^{s}\rho_{s}^{\alpha_{u}}du}e^{-\beta
c_{s}}ds$ $\displaystyle=-e^{-\beta(h_{0}x+Y_{0}^{i_{0}})}+\int_{0}^{t}\beta
e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}e^{-\beta(h_{s}X_{s}+Y_{s}^{\alpha_{s}})}\Big{[}-\frac{1}{\beta}e^{\beta(hX+Y)}e^{-\beta
c}-hc$ $\displaystyle\qquad\qquad\qquad-\Big{(}\frac{h}{\beta}(\ln
h-1)-h^{2}X-hY\Big{)}+h\Big{(}-\frac{1}{2}\beta
h|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(b^{i}-\beta\sigma^{i}z)\Big{)}-F^{i}(Z)\Big{]}ds$
$\displaystyle\qquad+\int_{0}^{t}\beta
e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}e^{-\beta(h_{s}X_{s}+Y_{s}^{\alpha_{s}})}(h\pi^{\prime}\sigma+Z^{\prime})dW$
$\displaystyle\qquad+\int_{0}^{t}e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}e^{-\beta
h_{s}X_{s}}\sum_{j,j^{\prime}\in\mathcal{M}}(e^{-\beta Y^{j}}-e^{-\beta
Y^{i}})I_{\\{\alpha_{s-}=j^{\prime}\\}}d\tilde{N}^{j^{\prime}j}_{s},$

where $(N^{j^{\prime}j})_{j^{\prime}j\in\mathcal{M}}$ are independent Poisson
processes each with intensity $q^{j^{\prime}j}$, and
$\tilde{N}_{t}^{j^{\prime}j}=N_{t}^{j^{\prime}j}-q^{j^{\prime}j}t,\ t\geqslant
0$ are the corresponding compensated Poisson martingales under the filtration
$\mathcal{F}$. And we drop the superscript $\alpha_{s}$ in the above integral.

Noting that for any $(\pi,c)\in\mathcal{U}$, the wealth process $X$ is
continuous, hence bounded on $[0,T]$ a.s. Therefore, the stochastic integrals
in the last equation are local martingales. Hence, there exists a sequence of
increasing stopping times $(\tau_{n})_{n\in\mathbb{N}}$ satisfying
$\lim_{n\rightarrow\infty}\tau_{n}=T$, a.s. such that

$\displaystyle\quad{\mathbb{E}}\Big{[}-e^{-\int_{0}^{\tau_{n}}\rho_{s}^{\alpha_{s}}ds}e^{-\beta(h_{\tau_{n}}X_{\tau_{n}}+Y_{\tau_{n}}^{\alpha_{\tau_{n}}})}-\int_{0}^{\tau_{n}}e^{-\int_{0}^{s}\rho_{s}^{\alpha_{u}}du}e^{-\beta
c_{s}}ds\Big{]}$
$\displaystyle=-e^{-\beta(h_{0}x+Y_{0}^{i_{0}})}+\int_{0}^{\tau_{n}}\beta
e^{-\int_{0}^{s}\rho_{u}^{\alpha_{u}}du}e^{-\beta(h_{s}X_{s}+Y_{s}^{\alpha_{s}})}\Big{[}-\frac{1}{\beta}e^{\beta(hX+Y)}e^{-\beta
c}-hc$ $\displaystyle\qquad\qquad\qquad-\Big{(}\frac{h}{\beta}(\ln
h-1)-h^{2}X-hY\Big{)}+h\Big{(}-\frac{1}{2}\beta
h|\pi^{\prime}\sigma^{i}|^{2}+\pi^{\prime}(b^{i}-\beta\sigma^{i}z)\Big{)}-F^{i}(Z)\Big{]}ds.$

Because $h,\beta>0$,

$\operatorname*{ess\;sup}_{c\in{\mathbb{R}}}\Big{[}-\frac{1}{\beta}e^{\beta(hX+Y)}e^{-\beta
c}-hc\Big{]}=\frac{h}{\beta}(\ln h-1)-h^{2}X-hY,$

we obtain from the definition of $F^{i}$ that

$\displaystyle{\mathbb{E}}\bigg{[}-e^{-\int_{0}^{\tau_{n}}\rho_{s}^{\alpha_{s}}ds}e^{-\beta(h_{\tau_{n}}X_{\tau_{n}}+Y_{\tau_{n}}^{\alpha_{\tau_{n}}})}\bigg{]}-{\mathbb{E}}\bigg{[}\int_{0}^{\tau_{n}}e^{-\int_{0}^{s}\rho_{s}^{\alpha_{u}}du}e^{-\beta
c_{s}}ds\bigg{]}\leqslant-e^{-\beta(h_{0}x+Y_{0}^{i_{0}})},$

for any $(c,\pi)\in\mathcal{U}$. As $n\to\infty$, the second expectation is
convergent by the monotone convergence theorem; while the first expectation,
because $(e^{-\beta h_{t}X_{t}^{\pi,c}})_{0\leqslant t\leqslant T}$ belongs to
class (D), $Y^{i}$, $i\in\mathcal{M}$, are bounded, is also convergent by the
dominated convergence theorem. Hence

$\displaystyle{\mathbb{E}}\bigg{[}-e^{-\int_{0}^{T}\rho_{s}^{\alpha_{s}}ds}e^{-\beta
X_{T}}-\int_{0}^{T}e^{-\int_{0}^{s}\rho_{s}^{\alpha_{u}}du}e^{-\beta
c_{s}}ds\bigg{]}\leqslant-e^{-\beta(h_{0}x+Y_{0}^{i_{0}})},$ (5.11)

by virtue of the terminal conditions $h_{T}=1$ and $Y_{T}=0$. Furthermore, the
inequality (5.11) becomes an equality for some $(\hat{\pi},\hat{c})$ if and
only if

$\displaystyle\hat{c}_{t}$
$\displaystyle=\operatorname*{argmax}_{c\in{\mathbb{R}}}\Big{[}-\frac{1}{\beta}e^{\beta(hX+Y^{\alpha})}e^{-\beta
c}-hc\Big{]}=-\frac{1}{\beta}\ln h+hX+Y^{\alpha},$

and

$\displaystyle\hat{\pi}_{t}\in\operatorname*{argmax}_{\pi\in\Pi}\Big{[}-\frac{1}{2}\beta
h|\pi^{\prime}\sigma^{\alpha}|^{2}+\pi^{\prime}(b^{\alpha}-\beta\sigma^{\alpha}Z^{\alpha})\Big{]}.$

The admissibility issue $(\hat{\pi},\hat{c})\in\mathcal{U}$ is covered in a
more general case; please refer to the proof of Theorem 5.8. $\Box$

###  5.2 Case II: Random interest rate

###### Assumption 5

For all $i\in\mathcal{M}$,

$\displaystyle\rho^{i}\in L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}).$

Furthermore, the processes $r,\ \mu,\ \sigma$ are independent of the regime
$i\in\mathcal{M}$ and such that

$\displaystyle r\in L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R}),\ \mu\in
L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}^{m}),\ \sigma\in
L_{\mathbb{F}^{W}}^{\infty}(0,T;\mathbb{R}^{m\times n}),$

and $\sigma\sigma^{\prime}\geqslant\delta I_{m}$ with some constant
$\delta>0$.

###### Assumption 6

$m=n$ and $\Pi=\mathbb{R}^{m}$, i.e. neither the portfolio nor the consumption
has constraints.

Under Assumptions 5 and 6, $\sigma$ is a square invertible matrix.

Consider the following BSDE system:

$\displaystyle\begin{cases}dh_{t}=-[rh-h^{2}-\eta^{\prime}\sigma^{-1}b-\frac{1}{h}|\eta|^{2}]dt+\eta^{\prime}dW,\\\
h_{T}=1.\end{cases}$ (5.12)

###### Theorem 5.4

Under Assumptions 5 and 6, (5.12) admits a unique solution $(h,\eta)\in
L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ such that
$h\geqslant\delta$ for some positive constant $\delta$.

Proof: Let $(p,q)\in L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ denote unique
solution to the following linear BSDE system:

$\displaystyle\begin{cases}dp_{t}=-[1-rp-q^{\prime}\sigma^{-1}b]dt+q^{\prime}dW,\\\
p_{T}=1.\end{cases}$ (5.13)

Indeed, $p$ admits the explicit representation:

$\displaystyle
p_{t}=\widetilde{{\mathbb{E}}}\bigg{[}e^{-\int_{t}^{T}r_{s}ds}+\int_{t}^{T}e^{-\int_{t}^{s}r_{u}du}ds\;\bigg{|}\;\mathcal{F}^{W}_{t}\bigg{]},$

where $\widetilde{{\mathbb{E}}}$ is the expectation with respect to the
probability measure $\widetilde{\mathbb{P}}$ defined by

$\frac{d\widetilde{\mathbb{P}}}{d\mathbb{P}}=\mathcal{E}\bigg{(}\int_{0}^{T}(\sigma^{-1}b)^{\prime}dW\bigg{)}.$

It follows from Assumption 5 that $\delta_{1}\leqslant p\leqslant\delta_{2}$
for some $0<\delta_{1}<\delta_{2}$. Therefore
$(h,\eta):=(\frac{1}{p},-\frac{q}{p^{2}})$ is well defined and $(h,\eta)\in
L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ with
$h\geqslant\frac{1}{\delta_{2}}>0$. It can be directly verified, using Itô’s
formula, that $(h,\eta)$ is a solution of (5.12). As for uniqueness, observe
that if $(h,\eta)$ is a solution of (5.12) with $h\geqslant\delta$ for some
$\delta>0$, then $(p,q^{i})=(\frac{1}{h},-\frac{\eta}{h^{2}})$ is a solution
to (5.13). Uniqueness follows from the fact that (5.13) has a unique solution.
$\Box$

Let $(h,\eta)$ denote the unique solution to (5.12). Consider the following
BSDE system:

$\displaystyle\begin{cases}dY^{i}=-\Big{[}-hY^{i}-(\sigma^{-1}b+\frac{\eta}{h})^{\prime}Z^{i}+\frac{1}{2\beta
h^{2}}|h\sigma^{-1}b+\eta|^{2}\\\
\qquad\qquad\quad+\frac{\rho^{i}}{\beta}-\frac{h}{\beta}(1-\ln
h)-\frac{1}{\beta}\sum_{j=1}^{\ell}q^{ij}e^{-\beta(Y^{j}-Y^{i})}\Big{]}dt+(Z^{i})^{\prime}dW,\\\
Y^{i}_{T}=0,\quad i\in\mathcal{M}.\end{cases}$ (5.14)

###### Remark 5.5

Because $\eta\in L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$
is not bounded, we cannot apply [8, Theorem 2.4] to solve (5.14).

Set
$(P^{i},\Lambda^{i})=(\frac{Y^{i}}{h},\frac{1}{h}Z^{i}-\frac{Y^{i}}{h^{2}}\eta)$
for all $i\in\mathcal{M}$. Then $(P^{i},\Lambda^{i})_{i\in\mathcal{M}}$
fulfills the following BSDE system:

$\displaystyle\begin{cases}dP^{i}=-\Big{[}-rP^{i}-(\sigma^{-1}b)^{\prime}\Lambda^{i}+\frac{1}{2\beta
h^{3}}|h\sigma^{-1}b+\eta|^{2}\\\ \qquad\qquad\quad+\frac{\rho^{i}}{\beta
h}-\frac{1}{\beta}(1-\ln h)-\frac{1}{\beta h}\sum_{j=1}^{\ell}q^{ij}e^{-\beta
h(P^{j}-P^{i})}\Big{]}dt+(\Lambda^{i})^{\prime}dW,\\\ P^{i}_{T}=0,\quad
i\in\mathcal{M}.\end{cases}$ (5.15)

###### Theorem 5.6

Under Assumptions 5 and 6, (5.15) admits a unique solution
$(P^{i},\Lambda^{i})_{i\in\mathcal{M}}$ such that $(P^{i},\Lambda^{i})\in
L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ for all
$i\in\mathcal{M}$.

Proof: We take the notations in the proof of Theorem 5.4. Under the
probability measure $\widetilde{\mathbb{P}}$,

$\widetilde{W}_{t}:=W_{t}+\int_{0}^{t}\sigma^{-1}bds,$

is an $n$-dimensional Brownian motion. Notice that $\eta\in
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$, hence
$\int_{0}^{t}\eta^{\prime}d\widetilde{W}_{s}$ is a BMO martingale under
$\widetilde{\mathbb{P}}$.

Let $k_{1}>0$ be a large constant such that, for all $i\in\mathcal{M}$ and
$t\in[0,T]$,

$\displaystyle-r\leqslant k_{1},\quad\frac{\rho^{i}}{\beta
h}-\frac{1}{\beta}(1-\ln h)\geqslant-k_{1},$ (5.16)

and

$\displaystyle\widetilde{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{-\int_{t}^{s}r_{u}du}\Big{(}\frac{1}{2\beta
h^{3}}|h\sigma^{-1}b+\eta|^{2}+\frac{\rho^{i}}{\beta h}-\frac{1}{\beta}(1-\ln
h)-\frac{1}{\beta
h}q^{ii}\Big{)}ds\;\bigg{|}\;\mathcal{F}^{W}_{t}\bigg{]}\leqslant k_{1}.$

Denote $k:=\max\\{e^{k_{1}T},k_{1}\\}$.

Define a truncating function $\psi:{\mathbb{R}}\rightarrow{\mathbb{R}}$ as

$\psi(y):=\max\\{-k,\min\\{y,k\\}\\}.$

Consider the following truncated system of (5.15),

$\displaystyle\begin{cases}dP^{i}=-\Big{[}-rP^{i}-(\sigma^{-1}b)^{\prime}\Lambda^{i}+\frac{1}{2\beta
h^{3}}|h\sigma^{-1}b+\eta|^{2}\\\ \qquad\qquad\quad+\frac{\rho^{i}}{\beta
h}-\frac{1}{\beta}(1-\ln h)-\frac{1}{\beta h}\sum_{j=1}^{\ell}q^{ij}e^{-\beta
h(\psi(P^{j})-\psi(P^{i}))}\Big{]}dt+(\Lambda^{i})^{\prime}dW,\\\
P^{i}_{T}=0,\quad i\in\mathcal{M}.\end{cases}$ (5.17)

The driver is Lipschitz continuous except for $\eta\in
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ is unbounded. By
[6, Theorem 9.3.5], there is a unique solution
$(P^{k,i},\Lambda^{k,i})_{i\in\mathcal{M}}$, such that
$(P^{k,i},\Lambda^{k,i})\in L^{2}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ for all $i\in\mathcal{M}$. If we
can show $|P^{k,i}|\leqslant k$ and $\Lambda^{k,i}\in
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$, then
$\psi(P^{k,i})=P^{k,i}$ and $(P^{k,i},\Lambda^{k,i})_{i\in\mathcal{M}}$ is a
solution to (5.15).

We first show that $P^{k,i}\geqslant-k$. It is direct to verify that

$\displaystyle(\underline{P},\underline{\Lambda})=(1-e^{k_{1}(T-t)},0),$

is the unique solution to the following BSDE:

$\displaystyle\begin{cases}d\underline{P}=-(k_{1}\underline{P}-k_{1})dt+\underline{\Lambda}^{\prime}dW\\\
\underline{P}_{T}=0.\end{cases}$ (5.18)

Notice that $\underline{P}\leqslant 0$ and

$\displaystyle k_{1}\underline{P}-k_{1}$
$\displaystyle\leqslant-r\underline{P}-(\sigma^{-1}b)^{\prime}\underline{\Lambda}+\frac{1}{2\beta
h^{3}}|h\sigma^{-1}b+\eta|^{2}$ $\displaystyle\qquad+\frac{\rho^{i}}{\beta
h}-\frac{1}{\beta}(1-\ln h)-\frac{1}{\beta h}\sum_{j=1}^{\ell}q^{ij}e^{-\beta
h(\psi(\underline{P})-\psi(\underline{P}))},$

so by Lemma 3.3

$-k\leqslant-e^{k_{1}T}\leqslant\underline{P}_{t}\leqslant P^{k,i}_{t}.$

We next show that $P^{k,i}\leqslant k$. From (5.17), we obtain

$\displaystyle e^{-\int_{0}^{t}r_{s}ds}P^{k,i}_{t}$
$\displaystyle=\int_{t}^{T}e^{-\int_{0}^{s}r_{u}du}\Big{[}-(\sigma^{-1}b)^{\prime}\Lambda^{k,i}+\frac{1}{2\beta
h^{3}}|h\sigma^{-1}b+\eta|^{2}$ $\displaystyle\qquad+\frac{\rho^{i}}{\beta
h}-\frac{1}{\beta}(1-\ln h)-\frac{1}{\beta h}\sum_{j=1}^{\ell}q^{ij}e^{-\beta
h(\psi(P^{k,j})-\psi(P^{k,i}))}\Big{]}ds-\int_{t}^{T}(\Lambda^{k,i})^{\prime}dW$
$\displaystyle=\int_{t}^{T}e^{-\int_{0}^{s}r_{u}du}\Big{[}\frac{1}{2\beta
h^{3}}|h\sigma^{-1}b+\eta|^{2}+\frac{\rho^{i}}{\beta h}-\frac{1}{\beta}(1-\ln
h)$ $\displaystyle\qquad-\frac{1}{\beta h}\sum_{j=1}^{\ell}q^{ij}e^{-\beta
h(\psi(P^{k,j})-\psi(P^{k,i}))}\Big{]}ds-\int_{t}^{T}(\Lambda^{k,i})^{\prime}d\widetilde{W}.$

Taking expectation $\widetilde{\mathbb{E}}$, and noting $q^{ij}\geqslant 0$
for $i\neq j$, we have

$\displaystyle P^{k,i}_{t}$
$\displaystyle=e^{\int_{0}^{t}r_{s}ds}\widetilde{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{-\int_{0}^{s}r_{u}du}\Big{(}\frac{1}{2\beta
h^{3}}|h\sigma^{-1}b+\eta|^{2}+\frac{\rho^{i}}{\beta h}-\frac{1}{\beta}(1-\ln
h)$ $\displaystyle\qquad\qquad\qquad-\frac{1}{\beta h}q^{ii}-\frac{1}{\beta
h}\sum_{j\neq i}^{\ell}q^{ij}e^{-\beta
h(\psi(P^{k,j})-\psi(P^{k,i}))}\Big{)}ds\;\bigg{|}\;\mathcal{F}^{W}_{t}\bigg{]}$
$\displaystyle\leqslant\widetilde{\mathbb{E}}^{i}\bigg{[}\int_{t}^{T}e^{-\int_{t}^{s}r_{u}du}\Big{(}\frac{1}{2\beta
h^{3}}|h\sigma^{-1}b+\eta|^{2}+\frac{\rho^{i}}{\beta h}-\frac{1}{\beta}(1-\ln
h)-\frac{1}{\beta h}q^{ii}\Big{)}ds\;\bigg{|}\;\mathcal{F}^{W}_{t}\bigg{]}$
$\displaystyle\leqslant k_{1}\leqslant k.$

Therefore $|P^{k,i}|\leqslant k$. Whence we can drop the superscript $k$ in
$(P^{k,i},\Lambda^{k,i})_{i\in\mathcal{M}}$.

Applying Itô’s formula to $(P^{i})^{2}$, we have for any stopping time
$\tau\leqslant T$,

$\displaystyle\quad(P^{i}_{\tau})^{2}+{\mathbb{E}}\bigg{[}\int_{\tau}^{T}|\Lambda^{i}|^{2}ds\;\bigg{|}\;\mathcal{F}^{W}_{\tau}\bigg{]}$
$\displaystyle={\mathbb{E}}\bigg{[}\int_{\tau}^{T}2P^{i}\Big{(}-rP^{i}-(\sigma^{-1}b)^{\prime}\Lambda^{i}+\frac{1}{2\beta
h^{3}}|h\sigma^{-1}b+\eta|^{2}$
$\displaystyle\qquad\qquad\quad+\frac{\rho^{i}}{\beta h}-\frac{1}{\beta}(1-\ln
h)-\frac{1}{\beta h}\sum_{j=1}^{\ell}q^{ij}e^{-\beta
h(P^{j}-P^{i})}\Big{)}ds\;\bigg{|}\;\mathcal{F}^{W}_{\tau}\bigg{]}$
$\displaystyle\leqslant{\mathbb{E}}\bigg{[}\int_{\tau}^{T}\Big{(}K+\frac{1}{2}|\Lambda^{i}|^{2}+K|\eta|^{2}\Big{)}ds\;\bigg{|}\;\mathcal{F}^{W}_{\tau}\bigg{]}.$

From which, we obtain

$\displaystyle{\mathbb{E}}\bigg{[}\int_{\tau}^{T}|\Lambda^{i}|^{2}ds\;\bigg{|}\;\mathcal{F}^{W}_{\tau}\bigg{]}\leqslant
K{\mathbb{E}}\bigg{[}\int_{\tau}^{T}\Big{(}1+|\eta|^{2}\Big{)}ds\;\bigg{|}\;\mathcal{F}^{W}_{\tau}\bigg{]}\leqslant
K,$

therefore $\Lambda^{i}\in
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$. The uniqueness
comes from [13, Lemma 3.4]. $\Box$

From Theorem 5.4 and Theorem 5.6, we immediately have

###### Theorem 5.7

Under Assumptions 5 and 6, (5.14) admits a unique solution
$(Y^{i},Z^{i})_{i\in\mathcal{M}}$ such that $(Y^{i},Z^{i})\in
L^{\infty}_{\mathcal{F}^{W}}(0,T;\mathbb{R})\times
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ for all
$i\in\mathcal{M}$.

###### Theorem 5.8

Suppose Assumptions 5 and 6 hold. Let $(h,\eta)$ and
$(Y^{i},Z^{i})_{i\in\mathcal{M}}$ be the unique solutions to (5.12) and (5.14)
respectively. Then the value function of the optimization problem (5.2) is
given by

$V(x,i_{0})=-e^{-\beta(h_{0}x+Y_{0}^{i_{0}})},$

and the optimal portfolio is

$\displaystyle\hat{\pi}_{t}=-\frac{1}{\beta
h^{2}_{t}}(\sigma^{\prime}_{t})^{-1}(\beta h_{t}\eta_{t}X_{t}+\beta
h_{t}Z_{t}^{\alpha_{t}}-h_{t}\sigma_{t}^{-1}b_{t}-\eta_{t}),$ (5.19)

the optimal consumption is

$\displaystyle\hat{c}_{t}=h_{t}X_{t}+Y_{t}^{\alpha_{t}}-\frac{1}{\beta}\ln
h_{t}.$ (5.20)

Proof: For any $(\pi,c)\in\mathcal{U}$, applying Itô’s formula to
$-e^{-\int_{0}^{t}\rho_{s}^{\alpha_{s}}ds}e^{-\beta(h_{t}X_{t}+Y_{t}^{\alpha_{t}})}$,
we obtain

$\displaystyle{\mathbb{E}}\Big{[}-e^{-\int_{0}^{\tau_{n}}\rho_{s}^{\alpha_{s}}ds}e^{-\beta(h_{\tau_{n}}X_{\tau_{n}}+Y_{\tau_{n}}^{\alpha_{\tau_{n}}})}\Big{]}-{\mathbb{E}}\Big{[}\int_{0}^{\tau_{n}}e^{-\int_{0}^{s}\rho_{s}^{\alpha_{u}}du}e^{-\beta
c_{s}}ds\Big{]}\leqslant-e^{-\beta(h_{0}x+Y_{0}^{i_{0}})},$

for a sequence of increasing stopping times $(\tau_{n})_{n\in\mathbb{N}}$
satisfying $\lim_{n\rightarrow\infty}\tau_{n}=T$. Same as the deterministic
case, the two expectations are both convergence as $n\to\infty$ so that

$\displaystyle{\mathbb{E}}\Big{[}-e^{-\int_{0}^{T}\rho_{s}^{\alpha_{s}}ds}e^{-\beta
X_{T}}\Big{]}-{\mathbb{E}}\Big{[}\int_{0}^{T}e^{-\int_{0}^{s}\rho_{s}^{\alpha_{u}}du}e^{-\beta
c_{s}}ds\Big{]}\leqslant-e^{-\beta(h_{0}x+Y_{0}^{i_{0}})},$ (5.21)

thanks to the terminal conditions $h_{T}=1$ and $Y_{T}=0$. Moreover, the
inequality (5.21) becomes an equality for some $(\hat{\pi},\hat{c})$ if and
only if (5.19) and (5.20) hold.

Denote $\hat{X}$ by the corresponding wealth process under the strategy
$(\hat{\pi},\hat{c})$. Then we have

$\displaystyle{\mathbb{E}}\Big{[}-e^{-\int_{0}^{\tau_{n}}\rho_{s}^{\alpha_{s}}ds}e^{-\beta(h_{\tau_{n}}^{\alpha_{\tau_{n}}}\hat{X}_{\tau_{n}}+Y_{\tau_{n}}^{\alpha_{\tau_{n}}})}-\int_{0}^{\tau_{n}}e^{-\int_{0}^{s}\rho_{s}^{\alpha_{u}}du}e^{-\beta\hat{c}_{s}}ds\Big{]}=-e^{-\beta(h_{0}x+Y_{0}^{i_{0}})}.$

Note that $\rho$, $Y^{i}$, $i\in\mathcal{M}$, are bounded processes,

$\displaystyle{\mathbb{E}}\Big{[}e^{-\beta
h_{\tau_{n}}^{\alpha_{\tau_{n}}}\hat{X}_{\tau_{n}}}+\int_{0}^{\tau_{n}}e^{-\beta\hat{c}_{s}}ds\Big{]}$
$\displaystyle\leqslant
K{\mathbb{E}}\Big{[}e^{-\int_{0}^{\tau_{n}}\rho_{s}^{\alpha_{s}}ds}e^{-\beta(h_{\tau_{n}}^{\alpha_{\tau_{n}}}\hat{X}_{\tau_{n}}+Y_{\tau_{n}}^{\alpha_{\tau_{n}}})}+\int_{0}^{\tau_{n}}e^{-\int_{0}^{s}\rho_{s}^{\alpha_{u}}du}e^{-\beta\hat{c}_{s}}ds\Big{]}$
$\displaystyle=Ke^{-\beta(h_{0}x+Y_{0}^{i_{0}})}.$

Applying Fatou’s lemma to the above, we obtain

$\displaystyle{\mathbb{E}}\Big{[}e^{-\beta\hat{X}_{T}}+\int_{0}^{T}e^{-\beta\hat{c}_{s}}ds\Big{]}\leqslant
K.$ (5.22)

Set $\theta:=\frac{1}{h}\sigma^{-1}(hb+\sigma\eta)$, then $\theta\in
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ and
$\mathcal{E}(\int_{0}^{t}\theta_{s}dW)$ is a uniformly integrable martingale.
By Girsanov’s theorem,

$W^{\mathbb{Q}}_{t}:=W_{t}+\int_{0}^{t}\theta_{s}ds$

is an $n$-dimensional Brownian motion under the measure $\mathbb{Q}$ defined
by

$\frac{d\mathbb{Q}}{d\mathbb{P}}=\mathcal{E}\Big{(}-\int_{0}^{T}\theta_{s}dW_{s}\Big{)}.$

We need to show that $(e^{-\beta h_{t}\hat{X}_{t}})_{0\leqslant t\leqslant T}$
belongs to class (D). Compared with Section 4 or [4], a new term $\eta\in
L^{2,\;\mathrm{BMO}}_{\mathcal{F}^{W}}(0,T;\mathbb{R}^{n})$ emerges (In
Section 4 or [4], $r$ is deterministic, hence $\eta=0$). Applying Itô’s
formula to $h_{t}\hat{X}_{t}$,

$\displaystyle h_{t}\hat{X}_{t}$
$\displaystyle=h_{0}x+\int_{0}^{t}\Big{[}-\frac{1}{\beta h^{2}}(\beta
hZ-h\sigma^{-1}b-\eta)^{\prime}\sigma^{-1}(hb+\sigma\eta)-h(Y-\frac{1}{\beta}\ln
h)\Big{]}ds$ $\displaystyle\qquad\quad-\int_{0}^{t}\frac{1}{\beta h}(\beta
hZ-h\sigma^{-1}b-\eta)^{\prime}dW$
$\displaystyle=h_{0}x+\int_{0}^{t}\Big{[}-h(Y-\frac{1}{\beta}\ln
h)\Big{]}ds-\int_{0}^{t}\frac{1}{\beta h}(\beta
hZ-h\sigma^{-1}b-\eta)^{\prime}dW^{\mathbb{Q}}$

Since $-\frac{1}{\beta h}(\beta hZ-h\sigma^{-1}b-\eta)\in
L^{2,\;\mathrm{BMO}}_{\mathcal{F}}(0,T;\mathbb{R}^{n})$, the process

$L_{t}:=-\int_{0}^{t}\frac{1}{\beta h}(\beta
hZ-h\sigma^{-1}b-\eta)^{\prime}dW^{\mathbb{Q}}$

is a BMO-martingale under $\mathbb{Q}$. Because $h$, $Y^{i}$,
$i\in\mathcal{M}$, are bounded, there exists a positive constant $K$ such that

$\displaystyle\frac{1}{K}e^{-\beta L_{t}}\leqslant e^{-\beta
h_{t}\hat{X}_{t}}\leqslant Ke^{-\beta L_{t}}\ \mbox{for all}\ t\in[0,T].$

Hence our problem reduces to show $(e^{-\beta L_{t}})_{0\leqslant t\leqslant
T}$ belongs to class (D). By the reverse Hölder inequality, there exists $p>1$
such that for all stopping times $\tau\leqslant T$,

$\displaystyle\Big{(}{\mathbb{E}}\Big{[}\mathcal{E}\Big{(}-\int_{0}^{T}\theta
dW\Big{)}^{p}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}\Big{)}^{\frac{1}{p}}\leqslant
K\mathcal{E}\Big{(}-\int_{0}^{\tau}\theta dW\Big{)}.$ (5.23)

Let $q$ be the conjugate exponent of $p$ i.e. $\frac{1}{p}+\frac{1}{q}=1$.
Since the function $x\mapsto e^{-\frac{\beta}{q}x}$ is convex, by Jensen’s
inequality, we have for any stopping time $\tau\leqslant T$,

$\displaystyle
e^{-\frac{\beta}{q}L_{\tau}}\leqslant{\mathbb{E}}^{\mathbb{Q}}\Big{[}e^{-\frac{\beta}{q}L_{T}}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}.$

It follows

$\displaystyle e^{-\beta L_{\tau}}$
$\displaystyle\leqslant\Big{(}{\mathbb{E}}^{\mathbb{Q}}\Big{[}e^{-\frac{\beta}{q}L_{T}}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}\Big{)}^{q}$
$\displaystyle=\Big{(}{\mathbb{E}}\Big{[}e^{-\frac{\beta}{q}L_{T}}\mathcal{E}\Big{(}-\int_{0}^{T}\theta
dW\Big{)}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}\Big{)}^{q}\mathcal{E}\Big{(}-\int_{0}^{\tau}\theta
dW\Big{)}^{-q}$ $\displaystyle\leqslant{\mathbb{E}}\Big{[}e^{-\beta
L_{T}}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}{\mathbb{E}}\Big{[}\mathcal{E}\Big{(}-\int_{0}^{T}\theta
dW\Big{)}^{p}\;\Big{|}\;\mathcal{F}_{\tau}\Big{]}^{\frac{q}{p}}\mathcal{E}\Big{(}-\int_{0}^{\tau}\theta
dW\Big{)}^{-q}$ $\displaystyle\leqslant K{\mathbb{E}}[e^{-\beta
L_{T}}\;\Big{|}\;\mathcal{F}_{\tau}],$

where we use Hölder’s inequality in the second inequality, (5.23) in the last
inequality. This shows that $(e^{-\beta h_{t}\hat{X}_{t}})_{0\leqslant
t\leqslant T}$ belongs to class (D) noting (5.22), hence the admissibility of
$(\hat{\pi},\hat{c})$. $\Box$

##  6 Concluding remarks

In this paper, we studied optimal consumption-investment problems in a market
with regime switching and random coefficients with coupled constraints.
Explicit solutions in term of the solutions of BSDE systems are given for the
cases of power, logarithmic and exponential utilities. Some of these BSDE
systems are new in the literature. Their solvability consists the main
mathematical contributions and is interesting in its own from the point of
view of BSDE theory. Extensions in other directions can be interesting as
well; for instance: (1) The problem in infinite time horizon (lifetime).
Actually the discount factor $\rho$ was introduced in problems (2.2) and (5.2)
for this possible extension. (2) The problem in an incomplete market, i.e.
$m<n$ in subsection 5.2 or when $r$, $\mu$, or $\sigma$ depends on the regime
$i$.

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