Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

- V. BALLY
- M.E. CABALLERO
**B. FERNANDEZ**

**Code(s) de Classification MSC:**

- 60G40 Stopping times; optimal stopping problems; gambling theory, See also {62L15, 90D60}
- 60J60 Diffusion processes, See also {58G32}
- 49J20 Optimal control problems involving partial differential equations
- 49L20 Dynamic programming method

**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.

**Mots Clés:** *Reflected Backward Stochastic Differential Equations; Variational
Inequalities; Optimal Stopping; Stochastic Control; Stochastic Flows*

**Date:** 1999-03-17

**Prépublication numéro:** *PMA-492*