Feynman-Kac formulas

$n\geq 1$, $d\geq 1$. Let ${\cal O}$ be an open subset of $(0,T)\times I\!\!R^n$. For $W$ a $d-$dimensional Brownian motion, we construct the process $X$ as the solution of the following SDE : :

(1)
\begin{align} \left\{ \begin{array}{l} dX^{t,x}_s=b(s,X_s^{t,x})ds+\sigma(s,X^{t,x}_s)dW_s,\mbox{ }\mbox{ }t\leq s\leq \tau \\ X^{t,x}_t=x \in \left\{y \left| (t,x) \in {\cal O}\right.\right\}, \end{array} \right. \end{align}

where $\tau=\inf\left\{s\in ]t,T] \left| (s,X_s^{t,x})\not\in{\cal O} \right.\right\}$ is the first exit time of $X$ from the domain $\cal O$.\
We then consider two other processes $Y$ and $Z$ defined by the following BSDE :

$$\left\{ \begin{array}{l} -dY^{t,x}_s=F(s,X_s^{t,x},Y_s^{t,x},Z_s^{t,x})ds-Z^{t,x}_sdW_s,\mbox{ }\mbox{ }t\leq s\leq \tau\Y^{t,x}_{\tau}=g(X^{t,x}_{\tau}).\\end{array} \right.$$
Writting $L\phi(t,x)=b(t,x)D\phi(t,x)+\frac{1}{2}\mbox{trace}\left[\sigma\sigma^*(t,x)D^2\phi(t,x)\right]$ for a regular function $\phi$ , we have, under some conditions
(see \cite{magdalena} p. 580), for every $t\leq s\leq \tau$ :

$$\left\{ \begin{array}{l} Y^{t,x}_s=u(s,X^{t,x}_s)\mbox{ }\Z^{t,x}_t=\sigma^*Du(s,X^{t,x}_s),\\end{array} \right.$$
where $u$ is the solution (eventually in a certain generalized sense) of the PDE :\
$$\left\{ \begin{array}{l} -\displaystyle\frac{\partial u}{\partial t}-Lu-F(t,x,u,\sigma^*(t,x)Du)=0\mbox{ in } {\cal O},\u(t,x)=g(t,x) \mbox{ on } \partial {\cal O}. \end{array} \right.$$

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