Représentation de Feynman–Kac dans des domaines temps–espace et sensibilité par rapport au domaine

We consider the solution X=(X_t)_t?0 of a time-inhomogeneous stochastic differential equation and the exit time τ by (t,X_t)_t?0 of the time–space domain $\text{D}$. We prove the differentiability of expectations of functionals of X stopped at τ, with respect to the domain $\text{D}$: these results extend those in the literature, known in particular by the analysts for the issues of shape optimization. However from the probabilistic point of view, this is not standard. To cite this article: C. Costantini et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).