The interrelation between stochastic differential inclusions and set-valued stochastic differential equations

In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L2L2 consisting of square integrable random vectors. We show that for the solution XX to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution xx for this inclusion that is a ‖⋅‖L2‖⋅‖L2-continuous selection of XX. This result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions, as well as to consider the viability problem for stochastic differential inclusions.