Stochastic control with exit time and constraints, application to small time attainability of sets (Q1879224)

The problem of stochastic viability before the exit time of a given open set is reduced to studying the properties of a suitably defined value function. In particular, this value function is characterized as the smallest nonnegative semicontinuous viscosity supersolution of a given Hamilton-Jacobi equation. The obtained results are applied to studying the small-time local attainability of a closed set with respect to the solutions of a controlled stochastic differential equation, and a sufficient condition is proved.