The viability theorem for stochastic differential inclusions

The aim of this paper is to combine two ways for representing uncertainty through stochastic differential inclusions: a "stochastic uncertainty", driven by a Wiener process, and a "contingent uncertainty", driven by a set-valued map, as well as to consider stochastic control problems with continuous dynamic and state dependent controls. This paper is also devoted to viability of a closed subset under stochastic differential inclusions, characterized in terms of stochastic tangent sets to closed subsets.