$$\begin{aligned}
\label{bornekm}
\theta_s \in [0,1]\,\,\, \mbox{ a.s. for all } t\leq s \leq T.
 \end{aligned}$$ We also impose the following non-bankruptcy constraint on the wealth process: $$\begin{aligned}
\label{contraintem}
X_s^{t,x,\theta} \geq  0 \,\,\mbox{ a.s. for all } t \leq s\leq T.\end{aligned}$$ Given an initial wealth $x\geq 0$ at time $t$, an admissible policy $\theta$ is a predictable stochastic process $(\theta_s)_{t\leq s\leq T}$ , such that conditions (\[bornekm\]) and (\[contraintem\]) are satisfied. We denote by $\Ac(t,x)$ the set of all admissible policies and $\Sc(t,x):=\{X^{t,x,\theta}\mbox{ such that } \theta\in \Ac(t,x)\}$.\
Our agent has preferences modeled by a utility function $U$.\
We assume that the agent’s utility is described by a CRRA utility function i.e. $U(x)=\frac{x^\eta}{\eta}$, where $\eta\in (0,1)$.\
We denote by $I$ the inverse of $U^{\prime}$ and we introduce the conjugate function of $U$ defined by $$\begin{aligned}
 \label{deftildeU}
\tilde U(y)&:=&\sup_{x>0}\{U(x)-xy\},\;\;\; y >0 \nonumber\\
&=&U(I(y))-yI(y).\end{aligned}$$ A straightforward calculus shows that $\displaystyle{\tilde U(y)=\frac{y^{-\gamma}}{\gamma}}$ where $\gamma=\frac{\eta}{1-\eta}$ and $\tilde U'(y) = -I(y)$ for all $ y >0$.\
The objective of the agent is to find the value function which is defined as $$\begin{aligned}
\label{pvaleur} 
v(t,x):=\Sup_{\theta \in \Ac(t,x)}E(U(X_T^{t,x,\theta})). \end{aligned}$$ [**Notations:**]{} The constants which appear in the paper are generic and could change from line to line.

Dual optimization 