Stochastic Control with Exit Time and Constraints, Application to Small Time Attainability of Sets

Abstract

We study the existence of a solution of controlled stochastic differential equations remaining in a given set of constraints at any time smaller than the exit time of a given open set. We also investigate the small time attainability of a given closed setK, i.e., the property that, for all arbitrary small time horizonT and for all initial condition in a sufficiently small neighborhood ofK, there exists a solution to the controlled stochastic differential equation which reachesK beforeT.