\section{Introduction}
Utility maximization represents an  important problems in financial mathematics. This is an optimal investment problem faced by an economic agent having the possibility of investing in a financial market over a finite period of time with fixed investment horizon $T$. The goal of the agent is to find an optimal portfolio that allows him to maximize his "welfare" at time $T$. The founding work of Von Neumann and Morgenstern \cite{VM44}, made it possible to represent the preferences of the investor by means of a function of utility U and a given probability measure P reflecting his views as follows:
$$E_{\mathbb{P}}[U(X_T^{x,\pi})]$$
Where $X_T^{x,\pi}$ is the wealth of investor at time $T$ starting from an initial wealth $x$ and adopting an investment strategy $\pi$.
The investor's problem then consists in solving the optimization
$$\sup_{\pi}E_{\mathbb{P}}[U(X_T^\pi)]$$
To solve this type of problem there are essentially two approaches:the dual approach \cite{SB09} and the BSDE approach \cite{IH2005}
\\
In reality, several scenarios are plausible and it is impossible to precisely identify $\mathbb{P}$. Therefore, we must take into account this ambiguity on the model also known as Knightian uncertainty. Knightian uncertainty studies have undergone enormous developments in both theory and applications. The work of  Maccheroni, Marinacci and Rustichini led to a new representation of preferences in the presence of model uncertainty, namely:
$$\inf_{Q \in \mathcal{Q}}E_{\mathbb{P}}[U(X_T^{x,\pi})+\gamma(Q)]$$
where $\mathcal{Q}$ the set of plausible senarios and $\gamma (Q)$ is the penalty term.In other words, the investor will decide in the worst case. Thus, it will solve the following optimization problem known as robust utility maximization problem:
$$\sup_{\pi}\inf_{Q \in \mathcal{Q}}E_{Q}[U(X_T^{x,\pi})+\gamma(Q)]$$
Several developments have been implemented on this subject, whether for the dual approach or the BSDE approach.
In this paper  we studies the robust utility maximization problem in incomplete market setting. We characterize the value function of our optimization problem using the quadratic BSDE. This was worked out through the class of nonlinear expectation called $g^*$- expectation.
\section{Formulation Problem}
We consider a filtred space $(\Omega, \mathcal{F}, \mathbb{F}^W, \mathbb{P})$ over a finite horizon time $T$, where the filtration $\mathbb{F}^W=(\mathcal{F}^W_t)_{t \in [0,T]}$ is generated by $d$- dimensional standard Brownian motion $W=(W^1,\ldots, W^d).$ 
For every measure $Q\ll P$ on $\mathcal{F}_T$ there is a predictable process $(\eta_t)_{t \in [0,T]}$
such that $\displaystyle E[\int_0^T\parallel \eta_t\parallel^2 dt]<+\infty \;\; Q.a.s$
 and the density process of $Q$ with respect to $\mathbb{P}$ is an RCLL
martingale $Z^Q =(Z^Q_t)_{t \in [0,T]}$ given by:
$$
Z_{t}^{Q}=\mathcal{E}\left(\int_{0}^{t} \eta_{u} d W_{u}\right) Q . a . s, \forall t \in[0, T]
$$
where $\mathcal{E}(M)_{t}=\exp \left(M_{t}-\frac{1}{2}\langle M\rangle_{t}\right)$ denotes the stochastic exponential of a continuous local martingale $M .$ We introduce a consistent time penalty given by:
$$
\gamma_{t}(Q)=E_{Q}\left[\int_{t}^{T} h\left(\eta_{s}\right) ds \mid \mathcal{F}_{t}\right]
$$
where $h: \mathbb{R}^{d} \rightarrow[0,+\infty]$ is a convex function, proper and lower semi-continuous function such that $h(0) \equiv 0 .$ We also assume that there are two positive constants $\kappa_{1}$ and $\kappa_{2}$ satisfying:
$$
h(x) \geq \kappa_{1}\|x\|^{2}-\kappa_{2}
$$
Our optimization problem  is written as follows:
$$\sup_{(\pi,c) \in\mathscr{A}_{e} }   \inf_{Q^\eta \in \mathcal{Q}}\mathbb{E}\Big[\bar{\alpha}U(X_T^{x,\pi,c})+\alpha\int_0^T e^{-\int_0^s \delta_udu} u(c_s)ds +\int_{0}^{T}e^{-\int_0^s \delta_udu} h\left(\eta_{s}\right) ds\Big]$$
Where  note that $\mathcal{Q}$ the space of all probability measures $Q^\eta $ on $(\Omega, \mathcal{F})$ such that $Q^\eta \ll \mathbb{P}$ on $\mathcal{F}_T$ and $\gamma_0(Q^\eta)< +\infty$. $U$ and $u$ are the utility functions and $\mathscr{A}_{e}$ is  the set of admissible strategies which will be specified later.
Our problem is then made up of two optimization subproblems. The first, the infmum, is studied by Faidi Matoussi and Mnif \cite{FMM17}. They have proven under the exponential integrability condition of the random variables $U(X_T^{x,\pi})$ and ${\displaystyle \int}_0^Tu(c_s)ds$ that the infimum is reached in a one unique probability measure $\mathbb{Q}^{*}$ which is equivalent  to $\mathbb{P}$ and and they characterized the value process of the dynamical optimization problem using the quadratic BSDE. More precisely, if
 $$\mathbb{E}_{\mathbb{P}}[\exp{\gamma |U(X_T^{x,\pi})|}]< +\infty \;\; \text{and} \;\; \mathbb{E}_{\mathbb{P}}[\exp{\gamma \int_0^T|u(c_s)|ds}]< +\infty \;\; \forall \gamma > 0$$ Then the process $$Y^{x,\pi,c}_t=\inf_{Q^\eta \in \mathcal{Q}}\mathbb{E}\Big[\bar{\alpha}U(X_T^{x,\pi,c})+\alpha\int_t^T e^{-\int_t^s \delta_udu} u(c_s)ds +\int_{t}^{T}e^{-\int_t^s \delta_udu} h\left(\eta_{s}\right) ds\Big]$$
 satisfied the following quadratic BSDE:
\begin{equation}\label{bsdecaraterization}
\left\{\begin{array}{l}
d Y_{t}^{x,\pi,c}=\left(\delta_{t} Y_{t}^{x,\pi,c}-\alpha u(c_t)+\beta h^{*}
(\frac{1}{\beta}Z_{t})\right) dt-Z_{t} d W_{t}, \\
Y_{T}^{x,\pi,c}=\bar{\alpha} \bar{U}(X_T^{x,\pi,c}) .
\end{array}\right.
\end{equation}

Where $h^*$ the Legendre-Fenchel transform of $h$.
\\
In the sequel we are interested in the second problem, the supermum.
\\
The financial market consists of one bond with interest rate zero and
$d \leq m$ stocks. In case $d<m$, we face an incomplete market. The price process
of stock $i$ evolves according to the equation:
$$
\begin{aligned}
&\frac{d S_{t}^{i}}{S_{t}^{i}}=b_{t}^{i} d t+\sigma_{t}^{i} d W_{t}, \quad i=1, \ldots, d
\end{aligned}
$$
where $b^{i}$ (resp. $\sigma^{i}$ ) is an $\mathbb{R}$ -valued (resp. $\mathbb{R}^{1 \times m}$ -valued) predictable uniformly bounded stochastic process. The lines of the $d \times m$ -matrix $\sigma$ are given by the vector $\sigma_{t}^{i}, i=1, \ldots, d$. The volatility matrix $\sigma=\left(\sigma^{i}\right)_{i=1, \ldots, d}$ has full rank and we assume that $\sigma \sigma^{t r}$ is uniformly elliptic, that is, $K I_{d} \geq \sigma \sigma^{t r} \geq \varepsilon I_{d}$ $P$ -a.s. for constants $K>\varepsilon>0 .$ The predictable $\mathbb{R}^{m}$ -valued process
$$
\theta_{t}=\sigma_{t}^{t r}\left(\sigma_{t} \sigma_{t}^{t r}\right)^{-1} b_{t}, \quad t \in[0, T]
$$
is then also uniformly bounded. A $d$ -dimensional $\mathbb{F}$ -predictable process $\pi=\left(\pi_{t}\right)_{0 \leq t \leq T}$ is called trading strategy if $\int \pi \frac{d S}{S}$ is well defined, for example, $\int_{0}^{T}\left\|\pi_{t} \sigma_{t}\right\|^{2} d t<\infty P$ -a.s. For $1 \leq i \leq d,$ the process $\pi_{t}^{i}$ describes the amount of money invested in stock $i$ at time $t .$ The number of shares is $\frac{\pi_{t}^{i}}{S_{t}^{2}} .$ The wealth process $X^{\pi}$ of a trading strategy $\pi$ with initial capital $x$ satisfies the equation
$$
X_{t}^{x,\pi}=x+\sum_{i=1}^{d} \int_{0}^{t}\pi_{i, u} \frac{d S_{i, u}}{S_{i, u}}-\int_0^t c_u du =x+\int_{0}^{t} \pi_{u} \sigma_{u}\left(d W_{u}+\theta_{u} d u\right)-\int_0^t c_u du, \quad t \in[0, T]
$$
In this notation $\pi$ has to be taken as a vector in $\mathbb{R}^{1 \times d}$. Trading strategies are self-financing. The investor uses his initial capital and during the trading interval $[0, T]$, there is no extra money flow out of or into his portfolio. Gains or losses are only obtained by trading with the stock.
\\
 Based on the above, the problem is equivalent to find:
\begin{equation}\label{opt}
V_0(x)=\sup_{(\pi,c) \in \mathscr{A}_{e}}Y_0^{x,\pi,c}
\end{equation}

such that
\begin{equation}
\left\{\begin{array}{l}
d Y_{t}^{x,\pi,c}=\left(\delta_{t} Y_{t}^{x,\pi,c}-\alpha u(c_t)+\beta h^{*}
(\frac{1}{\beta}Z_{t})\right) dt-Z_{t} d W_{t}, \\
Y_{T}^{x,\pi,c}=\bar{\alpha} \bar{U}(X_T^{x,\pi,c}) .
\end{array}\right.
\end{equation}

To solve this problem, we use the notion of $g$-expectation introduced by Peng \cite{Peng97} .
We then start with the main results on this notion of nonlinear expectation.
\section{ $g$-expectation}
The notion of $g$-expectations  was firstly introduced by Peng in \cite{Peng97} in the case of lipschitz generator.  from which most basic material of this section is taken 
Let $\xi$ a strictly positive random variable such that $\mathbb{E}_\mathbb{P}[\exp(\gamma |\xi|)]<\infty$ forall $\gamma>0$ and $g:\mathbb{R}\times \mathbb{R}^d \rightarrow \mathbb{R}$ satisfies the following conditions:
\begin{itemize}
\item $\forall  t \in [0, T ], g(t,0)=0$
\item $\forall t \in [0, T ], z \mapsto g(t, z)$ is a continuous convex function,
\item There are two positives constants $ \beta$ and $\gamma$ such that  $$ \forall (t,z)\in \mathbb{R}\times \mathbb{R}^d; |g(t,\ z)|\leq\beta|z|^{2}+\gamma .$$
\end{itemize}
 then  the BSDE
$$Y_{t}=\displaystyle \xi+\int_{t}^{T}g(s,\ Z_{s})ds-\int^{T}Z_{s}dW_{s},   
$$
admits   a  unique  solution  $(Y,\ Z)\in \mathscr{S}_T^{\infty}(\mathbb{R})\times \mathscr{H}_T^p\left(\mathbb{R}^m\right)$ where 
\begin{itemize}
\item[-] For $p \in \mathbb{N}, \mathscr{H}_T^p\left(\mathbb{R}^m\right)$ is the set of all $\mathbb{R}^m$-valued stochastic processes $Z$ which are predictable with respect to $\mathbb{F}$ and satisfy $\mathbb{E}\left[\left({\displaystyle \int}_0^T\left|Z_t\right|^2 d t\right)^{\frac{p}{2}}\right]<\infty$.
\item[-] $\mathscr{S}_T^{\infty}$ is the set of all continuous bounded $\mathbb{F}$-adapted semi-martingales  $Y$ that satisfy
$$
\forall \lambda>0, \quad \mathbb{E}\left[e^{\lambda \sup\limits_{t \in[0, T]}\left|Y_t\right|}\right]<\infty,
$$
\end{itemize}
$Y_{t}$ is called conditional $g$-expectation of $\xi$ under $\mathcal{F}_t$ and and is noted 
$\mathcal{E}_{g}[\xi|\mathcal{F}_t]$ and $Y_{0}$ is called $g$ -expectation of $\xi$, denoted by $Y_{0}=\mathcal{E}_{g}(\xi)$.

\begin{Definition}
\begin{enumerate}
\item
A process $X_{t}$ is called a $\mathrm{g}$-martingale, if for any $s \geq t$
$$\mathcal{E}_{g}[X_{t}|\mathcal{F}_s]=X_{s}$$
\item
A process $X_{t}$ is called a $\mathrm{g}$-submartingale, if for any $s \geq t$
$$\mathcal{E}_{g}[X_{t}|\mathcal{F}_s]\geq X_{s}$$
\item
A process $X_{t}$ is called a $\mathrm{g}$-supermartingale, if for any $s \geq t$
$$\mathcal{E}_{g}[X_{t}|\mathcal{F}_s]\leq X_{s}$$
\end{enumerate}
\end{Definition}
The following result is an immediate consequence of the comparison theorem
\begin{Proposition}
If $g_1 \leq g_2$ then any $\mathrm{g}_1$-supermartingale is $\mathrm{g}_2$-submartingale.
\end{Proposition}
\begin{Lemma}
$$Y_0^{x,\pi,c}={\cal E}_g[e^{-\int_0^T \delta_u du}\bar{\alpha} \bar{U}(X_T^{x,\pi,c})+\int_0^T\alpha e^{-\int_0^s \delta_u du} u(c_s)ds]$$
Where
$$g(w,t,z)=\beta e^{-\int_0^t \delta_u du} h^{*}
\left(\frac{1}{\beta}e^{\int_0^t \delta_u du}z\right)$$
\end{Lemma}
\begin{proof}
Let the stochastic process $L^{x,\pi,c}$ defined by:
$$L^{x,\pi,c}_t:= e^{-\int_0^t \delta_u du}Y^{x,\pi,c}_t+\int_0^t\alpha e^{-\int_0^s \delta_u du} u(c_s)ds$$
The process $L^{x,\pi,c}$ satisfy the following BSDE:
\begin{equation}\label{gcaracterization}
\left\{\begin{array}{l}
d L_{t}^{x,\pi,c}=\beta e^{-\int_0^t \delta_u du} h^{*}
\left(\frac{1}{\beta}e^{\int_0^t \delta_u du}Z_{t}\right) dt-Z_{t} d W_{t}, \\
L_{T}^{x,\pi,c}=e^{-\int_0^T \delta_u du}\bar{\alpha} \bar{U}(X_T^{x,\pi,c})+\int_0^T\alpha e^{-\int_0^s \delta_u du} u(c_s)ds .
\end{array}\right.
\end{equation}
In other words
$$L_{t}^{x,\pi,c}={\cal E}_g\Big[ e^{-\int_0^T \delta_u du}\bar{\alpha} \bar{U}(X_T^{x,\pi,c})+\int_0^T\alpha e^{-\int_0^s \delta_u du} u(c_s)ds |{\cal F}_t\Big] $$
\end{proof}
\begin{Remark}
The conditions mentioned above are verified by the BSDE(\ref{gcaracterization}). Indeed:
\begin{enumerate}
\item Since $h(0)=0$ and $h(x)\geq 0$ forall $x\in \mathbb{R},$ we have $h^*(0)=\sup\limits_{y \leq 0}-h(y)=0.$
\item $h$ is supposed to be continuous and convex function so $h^*$ is also.
\item Since $ \forall x \in \mathbb{R}^d; 
|h(x)| \geq \kappa_{1}\|x\|^{2}-\kappa_{2}
$
then 
$ \forall x \in \mathbb{R}^d; 
|h^*(x)| \leq \dfrac{1}{2\kappa_{1}}\|x\|^{2}-\kappa_{2}
$

\end{enumerate}
\end{Remark}
Thus our problem reduces to a utility maximization problem under the nonlinear expectation.

\section{Logarithme utility}

In this section, we consider the logarithm utility, which is given by
$$
U(x)=\ln(x)
$$
We here suppose that the initial wealth is strictly positive: $x > 0.$
We denote by $\pi_{t}$ the fraction of wealth invested in the risky assets at time $t \in[0, T],$ and $c_t$ the consumption rate at time $t$ that is, the portfolio value at time $t$  is given by
$$
\dfrac{dX_t^{\pi,c}}{X_t^{\pi,c}}=\pi_t\sigma_t(dW_t+\theta_tdt)-c_tdt\;\;; X_0^{\pi,c}=x
$$
and one can write
\begin{equation}\label{eq0}
X_t^{\pi,c}=x{\cal E}(\pi\sigma\textbf{.} W^{\mathbb{Q}})_t \exp(-\int_0^tc_sds)>0
\end{equation}
where ${\cal E}$ is the stochastic exponential and $W^{\mathbb{Q}}_t= W_t +\int_0^t \theta_sds.$
\\
To formulate consumption and investment constraints we introduce non-empty subsets ${\cal C} \in {\cal P}$ and ${\cal A} \in {\cal P}^{1\times m}$, where ${\cal P}$ denotes the set of all real-valued predictable processes $(c_t)_{0\ leq t \leq T}$ and ${\cal P}^{1\times m}$ the set of all predictable processes $(\pi_t)_{0\ leq t \leq T}$ with values in $\mathbb{R}^{1\times m}.$ 
\\
We assume that ${\cal C}$ and ${\cal A}$ are sequentially closed and ${\cal P}$-stable.
\begin{Definition}
an admissible strategy is a pair $(\pi,c) \in {\cal A} \times {\cal C}$  satisfying
\begin{equation}
\mathbb{E}[\exp(\gamma (|\ln(X_T^{x,\pi,c})|+\int_0^T |\ln(c_s)|ds)]< \infty ; \forall \gamma>0
\end{equation}
\begin{Proposition}
The set of admissible strategy is sequentially closed and ${\cal P}$-stable.
\end{Proposition}
\begin{proof}
let $(\pi^{(n)},c^{(n)})\in \mathscr{A}\times \mathscr{C}$ which converges $\nu \otimes \mathbb{P}$ a.e to $(\pi,c)$
\\
Since ${\cal C}$ and ${\cal A}$ are sequentially closed then $(\pi,c) \in {\cal A} \times {\cal C}.$
\\
By Fatou lemma, we have 
$$\mathbb{E}[\exp(\gamma (|\ln(X_T^{x,\pi,c})|+\int_0^T |\ln(c_s)|ds)] \leq \liminf\limits_{n}\mathbb{E}[\exp(\gamma (|\ln(X_T^{x,\pi^{(n)},c^{(n)}}|)+\int_0^T |\ln(c^{(n)}_s)|ds)]< \infty$$
and so $\mathscr{A}\times \mathscr{C}$ is sequentially closed.
\\
The ${\cal P}$-stability arises from convexity of the function $x\mapsto e^{|x|}.$

\end{proof}
\begin{Remark}
Using the same argument above, if ${\cal A}$ and ${\cal C}$ are ${\cal P}$-convex then $\mathscr{A}\times \mathscr{C}$ is ${\cal P}$-convex.
\end{Remark}
\end{Definition}
We assume in the sequel that there exists a pair $(\bar{\pi},\bar{c}) \in \mathscr{A}\times \mathscr{C}$ such that $\ln(c)-c$ and $\pi$ are bounded.
\\
To solve this problem we adopt the same approach as imkeller hu \cite{IH2005} and Jiang \cite{Jiang16}. 
\\Let us first start with the following useful lemma

\begin{Lemma}\label{lemma1}
For any given  $(\pi,c) \in  \mathscr{A}_{e}$ , if there exists a RCLL adapted process $R^{x,\pi,c}$  such that 
 \begin{itemize}
 \item $\forall (\pi,c) \in  \mathscr{A}\times \mathscr{C}  ; R_{T}^{x,\pi,c}:=\bar{\alpha}e^{-\int_0^T \delta_u du }\ln( X_{T}^{x,\pi,c})+{\displaystyle \int}_0^T \alpha e^{-\int_0^u \delta_s ds } \ln (c_u X_{u}^{x,\pi,c})du$ 
 \item $\forall (\pi,c) \in  \mathscr{A}\times \mathscr{C}  ; R^{x,\pi,c}$ is a $ g$-supermartingale;
 \item  There exists $(\pi^{*},c^{*}) \in \mathscr{A}\times \mathscr{C}$ such that $R^{ \pi^{*},c^*}$ is a $g$ -martingale,
 then $(\pi^{*},c^{*})$ is an optimal strategy to Problem (\ref{opt})
 \end{itemize}
\end{Lemma}
 \begin{proof}
 Proof. For given $t \in[0, T]$ and $(\hat{\pi},\hat{c}) \in \mathscr{A}_{e}$, we introduce $\tilde{\pi}_{u}=\pi_{u} I_{u \leq t}+\hat{\pi}_{u} I_{u>t}$, and $\tilde{c}_{u}=c_{u} I_{u \leq t}+\hat{c}_{u} I_{u>t}$ one can see that ($\tilde{\pi},\tilde{c}) \in \mathscr{A}_{e}$ and $X_{t}^{x,\tilde{\pi},\tilde{c}}=X_{t}^{x,\pi,c}$. Since $R^{e, \tilde{\pi}}$ is a $g$-supermartingale, we have
 \begin{align*}
 R_{t}^{x,\pi,c} & \geq {\cal E}_g[\bar{\alpha}e^{-\int_0^T \delta_u du }\Big(\ln( X_{T}^{x,\tilde{\pi},\tilde{c}})\Big)+{\displaystyle \int}_0^T \alpha e^{-\int_0^u \delta_s ds } \ln (\tilde{c}_u X_{u}^{x,\tilde{\pi},\tilde{c}})du|{\cal F}_t]
 \\&={\cal E}_g\Big[\bar{\alpha}e^{-\int_0^T \delta_u du }\Big(\ln( X_{t}^{x,\pi,c}\dfrac{X_{T}^{x,\hat{\pi},\hat{c}}}{X_{t}^{x,\hat{\pi},\hat{c}}})\Big)+{\displaystyle \int}_0^t \alpha e^{-\int_0^u \delta_s ds } \ln (c_u X_{u}^{x,\pi,c})du \\&
 +{\displaystyle \int}_t^T \alpha e^{-\int_0^u \delta_s ds } \ln (\hat{c}_u  X_{t}^{x,\pi,c}\dfrac{X_{u}^{x,\hat{\pi},\hat{c}}}{X_{t}^{x,\hat{\pi},\hat{c}}})du|{\cal F}_t\Big]
 \end{align*}
 thus,
 \begin{align*}
 R_{t}^{x,\pi,c} & \geq \sup\limits_{(\hat{\pi},\hat{c})}{\cal E}_g\Big[\bar{\alpha}e^{-\int_0^T \delta_u du }\ln( X_{t}^{x,\pi,c}\dfrac{X_{T}^{x,\hat{\pi},\hat{c}}}{X_{t}^{x,\hat{\pi},\hat{c}}})+{\displaystyle \int}_0^t \alpha e^{-\int_0^u \delta_s ds } \ln (c_u X_{u}^{x,\pi,c})du \\&
 +{\displaystyle \int}_t^T \alpha e^{-\int_0^u \delta_s ds } \ln (\hat{c}_u  X_{t}^{x,\pi,c}\dfrac{X_{u}^{x,\hat{\pi},\hat{c}}}{X_{t}^{x,\hat{\pi},\hat{c}}})du|{\cal F}_t\Big]
 \end{align*}
 Furthermore, as for $(\pi^*,c^*); $ $R_{t}^{ \pi^{*}}$ is a $g$ -martingale,
\begin{align*}
 R_{t}^{x,\pi^*,c^*} & = {\cal E}_g\Big[\bar{\alpha}e^{-\int_0^T \delta_u du }\ln( X_{t}^{x,\pi^*,c^*}\dfrac{X_{T}^{x,\pi^*,c^*}}{X_{t}^{x,\pi^*,c^*}})+{\displaystyle \int}_0^t \alpha e^{-\int_0^u \delta_s ds } \ln (c_u X_{u}^{x,\pi^*,c^*})du \\&
 +{\displaystyle \int}_t^T \alpha e^{-\int_0^u \delta_s ds } \ln (c^*_u  X_{t}^{x,\pi^*,c^*}\dfrac{X_{u}^{x,\pi^*,c^*}}{X_{t}^{x,\pi^*,c^*}})du|{\cal F}_t\Big]
 \\ & \leq
  \sup\limits_{(\hat{\pi},\hat{c})}{\cal E}_g\Big[\bar{\alpha}e^{-\int_0^T \delta_u du }\ln( X_{t}^{x,\pi^*,c^*}\dfrac{X_{T}^{x,\hat{\pi},\hat{c}}}{X_{t}^{x,\hat{\pi},\hat{c}}})+{\displaystyle \int}_0^t \alpha e^{-\int_0^u \delta_s ds } \ln (c_u X_{u}^{x,\pi^*,c^*})du \\&
 +{\displaystyle \int}_t^T \alpha e^{-\int_0^u \delta_s ds } \ln (\hat{c}_u  X_{t}^{x,\pi^*,c^*}\dfrac{X_{u}^{x,\hat{\pi},\hat{c}}}{X_{t}^{x,\hat{\pi},\hat{c}}})du|{\cal F}_t\Big]
 \end{align*}
 Thus
 \begin{align*}
 R_{t}^{x,\pi^*,c^*} & = 
  \sup\limits_{(\hat{\pi},\hat{c})}{\cal E}_g\Big[\bar{\alpha}e^{-\int_0^T \delta_u du }\ln( X_{t}^{x,\pi^*,c^*}\dfrac{X_{T}^{x,\hat{\pi},\hat{c}}}{X_{t}^{x,\hat{\pi},\hat{c}}})+{\displaystyle \int}_0^t \alpha e^{-\int_0^u \delta_s ds } \ln (c_u X_{u}^{x,\pi^*,c^*})du \\&
 +{\displaystyle \int}_t^T \alpha e^{-\int_0^u \delta_s ds } \ln (\hat{c}_u  X_{t}^{x,\pi^*,c^*}\dfrac{X_{u}^{x,\hat{\pi},\hat{c}}}{X_{t}^{x,\hat{\pi},\hat{c}}})du|{\cal F}_t\Big]
 \end{align*}
 holds for each $t \in [0, T]$, and by taking $t = 0$, we know that $(\pi^*,c^*)$
 is an optimal strategy
 \end{proof}
We are now trying to build a family of processes $R^{x,\pi,c}$  which satisfies the conditions of the previous lemma. We assume in the sequel that the process $\delta$ is deterministic. 
\\
We seek $R^{x,\pi,c}$ in the form
$$R_{t}^{x,\pi,c}:=\bar{\alpha}h(t)e^{-\int_0^t \delta_u du }\Big(\ln( X_{t}^{x,\pi,c})-Y_t\Big)+{\displaystyle \int}_0^t \alpha e^{-\int_0^u \delta_s ds } \ln (c_u X_{u}^{x,\pi,c})du$$ 
Where the process $Y$  satisfie a quadratic BSDE of the form
$$dY_t=\rho_t Y_t dt + f(t,Z_t)dt -Z_t dW_t\;\;\; Y_T=0$$
and $h$ is the solution of the ordinary Cauchy problem
\begin{equation}\label{cauchypb}
\forall 0 \leq t \leq T; \bar{\alpha} h'(t)=\bar{\alpha}\delta_t h(t)-\alpha \;\; \text{and} \;\; h(T)=1.
\end{equation}
So that
$$h(t)=e^{-\int_t^T \delta_u du }+\dfrac{\alpha}{\bar{\alpha}} e^{\int_0^t \delta_u du }\int_t^Te^{-\int_0^u \delta_s ds }du$$
Our goal now is to find the process $\rho$ and the function $f$ such that the process $R$ satisfies the conditions mentioned in the lemma (\ref{lemma1}).
\\ 
Using ito formula we have
\begin{align*}
dR_{t}^{x,\pi,c}&=\bar{\alpha}e^{-\int_0^t \delta_u du }(h'(t)-\delta_t h(t))\Big(\ln( X_{t}^{x,\pi,c})-Y_t\Big)dt
\\&+\bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(\pi_t \sigma_{t}dW_{t}-\pi_t \sigma_{t}\theta_{t}dt - c_tdt-\rho_t Y_t dt - f(t,Z_t)dt +Z_t dW_t)
\\&+\alpha e^{+\int_0^u \delta_s ds } \ln (c_t) dt +\alpha e^{-\int_0^u \delta_s ds } \ln (X_{t}^{x,\pi,c}) dt
\\&=-\alpha e^{-\int_0^t \delta_u du }\Big(\ln( X_{t}^{x,\pi,c})-Y_t\Big)dt
\\&+\bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(\pi_t \sigma_{t}dW_{t}-\pi_t \sigma_{t}\theta_{t}dt - c_tdt-\rho_t Y_t dt - f(t,Z_t)dt +Z_t dW_t)
\\&+\alpha e^{-\int_0^u \delta_s ds } \ln (c_t) dt +\alpha e^{-\int_0^u \delta_s ds } \ln (X_{t}^{x,\pi,c}) dt
\end{align*}
To get rid of the term $Y_t dt$, we choose $\rho_t=\dfrac{\alpha}{\bar{\alpha}h(t)} \forall 0\leq t \leq T.$ We obtain then
\begin{align*}
dR_{t}^{x,\pi,c}&= e^{-\int_0^t \delta_u du }\Big(\alpha  \ln (c_t) dt -\bar{\alpha}h(t)(\pi_t \sigma_{t}\theta_{t} + c_t+ f(t,Z_t))dt\Big)
+\bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(\pi_t \sigma_{t}+Z_t)dW_{t}
\\&=e^{-\int_0^t \delta_u du }\Big(\alpha  \ln (c_t)  -\bar{\alpha}h(t)c_t\Big)dt 
-\bar{\alpha}h(t) e^{-\int_0^t \delta_u du }\Big(\pi_t \sigma_{t}\theta_{t} + f(t,\dfrac{e^{\int_0^t \delta_u du }}{\bar{\alpha}h(t)}\tilde{Z}_t-\pi_t\sigma_t)\Big)dt
+\tilde{Z}_tdW_{t}
\end{align*}
Therefore $R^{x,\pi,c}$ is solution of the BSDE
$$
\left\{\begin{array}{l}
d R_{t}^{x,\pi,c}=F(t,Z^{\pi,c}_t)dt+Z^{\pi,c}_tdW_{t} 
 \\
R_{T}^{x,\pi,c}:=\bar{\alpha}e^{-\int_0^T \delta_u du }\ln( X_{T}^{x,\pi,c})+{\displaystyle \int}_0^T \alpha e^{-\int_0^u \delta_s ds } \ln (c_u X_{u}^{x,\pi,c})du.
\end{array}\right.
$$
Where
$$F(t,z)=e^{-\int_0^t \delta_u du }\Big(\alpha  \ln (c_t)  -\bar{\alpha}h(t)c_t\Big)
-\bar{\alpha}h(t) e^{-\int_0^t \delta_u du }\Big(\pi_t \sigma_{t}\theta_{t} + f(t,\dfrac{e^{\int_0^t \delta_u du }}{\bar{\alpha}h(t)}z-\pi_t\sigma_t)\Big)$$
we can choose $f$ such that:
$$ F(t,Z_t^{\pi,c}) \leq g \left(t, Z_t^{\pi,c}\right) \forall (\pi,c); P.a.s$$
with the equality is rached for some $(\pi^*,c^*) \in \mathscr{A} \times \mathscr{C}$
\\
It is sufficient that forall $z\in \mathbb{R}^d$; forall $(\pi,c)\in \mathscr{A}\times \mathscr{C}$ we have
\begin{equation}\label{eq1}
e^{-\int_0^t \delta_u du }\Big(\alpha  \ln (c_t)  -\bar{\alpha}h(t)c_t\Big)
-\bar{\alpha}h(t) e^{-\int_0^t \delta_u du }\Big(\pi_t \sigma_{t}\theta_{t} + f(t,\dfrac{e^{\int_0^t \delta_u du }}{\bar{\alpha}h(t)}z-\pi_t\sigma_t)\Big)
 \leq g \left(t, z\right)\;\;\;  \mathbb{P}.a.s
\end{equation}

and there is at least $(\pi^*,c^*)$ such that forall $z\in \mathbb{R}^d$
\begin{equation}\label{eq2}
e^{-\int_0^t \delta_u du }\Big(\alpha  \ln (c^*_t)  -\bar{\alpha}h(t)c^*_t\Big) 
-\bar{\alpha}h(t) e^{-\int_0^t \delta_u du }\Big(\pi^*_t \sigma_{t}\theta_{t} - f(t,\pi^*_t\sigma_t-\dfrac{e^{\int_0^t \delta_u du }}{\bar{\alpha}h(t)}z)\Big)
 = g \left(t, z\right)\;\;\;  \mathbb{P}.a.s
\end{equation}
Equation (\ref{eq1} ) implies that forall $z\in \mathbb{R}^d$; forall $\pi\in \mathscr{A}$ we have
\begin{equation}\label{eq3}
e^{-\int_0^t \delta_u du }\sup_{c\in \mathscr{C}}\Big(\alpha  \ln (c_t)  -\bar{\alpha}h(t)c_t\Big) 
-\bar{\alpha}h(t) e^{-\int_0^t \delta_u du }\Big(\pi_t \sigma_{t}\theta_{t} + f(t,\dfrac{e^{\int_0^t \delta_u du }}{\bar{\alpha}h(t)}z-\pi_t\sigma_t)\Big)
 \leq g \left(t, z\right)\;\;\;  \mathbb{P}.a.s
\end{equation}
which is equivalent to
$$\forall z, \forall \pi \in \mathscr{A}; f(t,z) \geq \frac{1}{\bar{\alpha}h(t)} \left(e^{-\int_0^t \delta_u du }\sup_{c\in \mathscr{C}}\Big(\alpha  \ln (c_t)  -\bar{\alpha}h(t)c_t\Big)- g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi_t\sigma_t)\right)\right)-\pi_t \sigma_{t} \theta_{t}\;\;\;  \mathbb{P}.a.s$$
and so
$$\forall z ; f(t,z) \geq \frac{1}{\bar{\alpha}h(t)} e^{-\int_0^t \delta_u du }\sup_{c\in \mathscr{C}}\Big(\alpha  \ln (c_t)  -\bar{\alpha}h(t)c_t\Big)- \inf_{\pi \in \mathscr{A}}\left(\frac{1}{\bar{\alpha}h(t)}g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi_t\sigma_t)\right)+\pi_t \sigma_{t} \theta_{t}\right)\;\;\;  \mathbb{P}.a.s$$
The sets $$\{c \in \mathscr{C}; \alpha  \ln (c)  -\bar{\alpha}h(t)c \geq \alpha  \ln (\bar{c})  -\bar{\alpha}h(t)\bar{c}\}$$  and $$\{\pi \in \mathscr{A}; g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi\sigma)\right)+\pi \sigma \theta \leq g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \bar{\pi}\sigma)\right)+\bar{\pi} \sigma \theta\}$$
are $L^0-$ bounded. It follows from Theorem \textbf{4.4} in \cite{cheridito11} the existence of a pair $(\pi^*,c^*) \in \mathscr{A} \times \mathscr{C}$ such that
$$\inf_{\pi \in \mathscr{A}}g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi_t\sigma_t)\right)+\pi_t \sigma_{t} \theta_{t}=g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi^*_t\sigma_t)\right)+\pi^*_t \sigma_{t} \theta_{t}$$
and
$$\sup_{c\in \mathscr{C}}\Big(\alpha  \ln (c_t)  -\bar{\alpha}h(t)c_t\Big)=\alpha  \ln (c^*_t)  -\bar{\alpha}h(t)c^*_t$$

To achieve equation (\ref{eq2}) we must choose
\begin{equation}\label{eq4}
\begin{split}
 f(t,z)& =
\frac{1}{\bar{\alpha}h(t)} e^{-\int_0^t \delta_u du }\sup_{c\in \mathscr{C}}\Big(\alpha  \ln (c_t)  -\bar{\alpha}h(t)c_t\Big)- \inf_{\pi \in \mathscr{A}}\left(\frac{1}{\bar{\alpha}h(t)}g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi_t\sigma_t)\right)+\pi_t \sigma_{t} \theta_{t}\right)
\\
& = 
 \frac{1}{\bar{\alpha}h(t)} \left(e^{-\int_0^t \delta_u du }\Big(\alpha  \ln (c^*_t)  -\bar{\alpha}h(t)c^*_t\Big)- g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi^*_t\sigma_t)\right)-\pi^*_t \sigma_{t} \theta_{t}\right)\;\;\;  \mathbb{P}.a.s
 \end{split}
\end{equation}
It remains to verify the measurability of the function $f$. We have the following lemma
\begin{Lemma}
\begin{enumerate}
\item $f$ is is predictible process
\item There are a predictables processes  $c^*$ and $\pi^*$ such that
\begin{equation*}
 f(t,z)=
 \frac{1}{\bar{\alpha}h(t)} \left(e^{-\int_0^t \delta_u du }\Big(\alpha  \ln (c^*_t)  -\bar{\alpha}h(t)c^*_t\Big)- g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi^*_t\sigma_t)\right)-\pi^*_t \sigma_{t} \theta_{t}\right)\;\;\;  \mathbb{P}.a.s
\end{equation*}
\end{enumerate}
\end{Lemma}
From the above we deduce the following result
\begin{Theorem}
The optimal value of the problem (\ref{opt}) is given by
$$V_0(x)=\bar{\alpha}h(0)\Big(\ln(x)-Y_0\Big)$$
Where $Y_.$ is the solution of the following BSDE
$$dY_t=\dfrac{\alpha}{\bar{\alpha}h(t)} Y_t dt + f(t,Z_t)dt-Z_t dW_t\;\;;\; Y_T=0$$
The functions $h$ and  is respectively  given  by (\ref{cauchypb}) and (\ref{eq4}).
\\
Morover $(\pi^*,c^*)$  is an optimal admissible strategy if and only if
$$c^* \in \argmax_{c\in {\cal C}} \Big(\alpha  \ln (c_t)  -\bar{\alpha}h(t)c_t\Big) \;\;\; \text{and}  \;\;\;\pi^* \in \argmin_{\pi\in {\cal A}} g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi_t\sigma_t)\right)+\pi_t \sigma_{t} \theta_{t}$$
\end{Theorem}
\section{Example}

In this section we suppose that $h(x)=\dfrac{1}{2}\|x\|^2; \forall x \in \mathbb{R}^d$ which matches the entropic utility case. Then, we have $  h^*(x)=\dfrac{1}{2}\|x\|^2$ and therefore
$g(w,t,z)=\dfrac{1}{\beta }e^{\int_0^t \delta_u du} \|z\|^2$.
We also assume that $\dfrac{\alpha}{\bar{\alpha}h} \in {\cal C}$
\begin{Proposition}
The generator (\ref{eq4}) is given by
\begin{equation*}
 f(t,z) =
\frac{1}{\bar{\alpha}h(t)} e^{-\int_0^t \delta_u du }(\alpha\ln(\dfrac{\alpha}{\bar{\alpha}h(t)})-\alpha)- \dfrac{\bar{\alpha}}{\beta}h(t)e^{-\int_0^t \delta_u du }d^2(z+\dfrac{\beta e^{\int_0^t \delta_u du } }{2\bar{\alpha} h(t) }\theta_t,\sigma {\cal A})-z_t\theta_t-\dfrac{\beta e^{\int_0^t \delta_u du } }{4\bar{\alpha} h(t) }\|\theta_t\|^2
\end{equation*}
\end{Proposition}
\begin{proof}
\begin{equation*}
\begin{split}
 &\frac{1}{\bar{\alpha}h(t)}g \left(t, \bar{\alpha}h(t)e^{-\int_0^t \delta_u du }(z+ \pi_t\sigma_t)\right)+\pi_t \sigma_{t} \theta_{t}
 \\&
 =\dfrac{\bar{\alpha}}{\beta}h(t)e^{-\int_0^t \delta_u du }\|z+\pi_t\sigma_t\|^2+\pi_t\sigma_t\theta_t
 \\&
 =\dfrac{\bar{\alpha}}{\beta}h(t)e^{-\int_0^t \delta_u du }\|z+\pi_t\sigma_t+\dfrac{\beta e^{\int_0^t \delta_u du } }{2\bar{\alpha} h(t) }\theta_t\|^2-z_t\theta_t-\dfrac{\beta e^{\int_0^t \delta_u du } }{4\bar{\alpha} h(t) }\|\theta_t\|^2
 \end{split}
\end{equation*}
\end{proof}


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