Stochastic differential inclusions in terms of infinitesimal generators and mean derivatives

Abstract

We look for a stochastic process ξ(t), generally speaking, on a finite-dimensional manifold, such that for its infinitesimal generator G(t, x) the inclusion G(t, ξ(t)) ∈ L(t, ξ(t)) holds a.s. where L(t, x) is a set-valued field of second-order vectors. We reduce this problem to some problems with the so-called mean derivatives that are investigated by involving the theory of connections on manifolds. The existence theorem on a manifold is proved on the basis of a technical result that gives conditions in terms of infinitesimal generators for weak compactness of measures on path spaces corresponding to solutions of stochastic differential equations. In a particular case of linear spaces we also prove an existence theorem of another sort formulated in terms of Itô type estimates.

Keywords:

• stochastic differential inclusions
• infinitesimal generators
• mean derivatives

Acknowledgment

The research is supported in part by RFBR Grants 07–01–00137 and 08–01–00155.