Dualité convexe,temps d'arrêt optimal et contrôle stochastique

The purpose of this paper is to apply convex analysis methods to prove existence results for optimal stopping time problems for the diffusion processes considered by Stroock and Varadhan. The work is based on the characterization given by Rost of the measures associated to the stopping times.The method is applied to the problem of control of diffusions where the stopping time is also a control.