Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Reflected BSDE's , PDE's and Variational Inequalities

Auteur(s):

Code(s) de Classification MSC:

• 60G40 Stopping times; optimal stopping problems; gambling theory, See also {62L15, 90D60}
Résumé: We discuss a class of semilinear PDE's with obstacle, of the form $(\partial _{t}+L)u+f(t,x,u,\sigma ^{*}\nabla u)+\mu =0,\quad u\geq h,u_{T}=g$ where $h$ is the obstacle. The solution of such an equation (in variational sense) is a couple $(u,\mu )$ where $u\in L^{2}([0,T];H^{1})$ and $\mu$ is a positive Radon measure concentrated on $\{u=h\}$. We prove that this equation has a unique solution and $u$ is the maximal solution of the corresponding variational inequality. The probabilistic interpretation (Feynman-Kac formula) is given by means of Reflected Backward Stochastic Differential Equations. Moreover $u$ is the value function of a mixed stochastic control problem and we use RBSDE's in order to produce an optimal stopping time and an optimal control. Problems with two barriers are also discussed.