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