Viability of moving sets for stochastic differential equation

Abstract

We study the existence of solutions of stochastic differential equations with a state constraint depending on the time. We provide a necessary and sufficient characterization of closed, time depending constraints for which there exists a solution of a given stochastic differential equation. This characterization is given in terms of viscosity super- and subsolution of some suitable partial differentiable equations. The above property, called viability, is stated for both forward and backward stochastic differential equations.