This chapter is devoted to the theory of stochastic differential inclusions. The main results deal with stochastic functional inclusions defined by set-valued functional stochastic integrals. Subsequent sections discuss properties of stochastic and backward stochastic differential inclusions.
Stochastic Functional InclusionsThroughout this section, by P F D . ; F ; F; P / we shall denote a complete filtered probability space and assume thatd m / satisfy the following conditions .H/:(i) F and G are measurable, (ii) F and G are uniformly square integrably bounded.For set-valued mappings F and G as given above, by stochastic functional inclusions SF I.F; G/, SF I.F ; G/, and SF I .F; G/ we mean relations of the formrespectively, which have to be satisfied for every 0 Ä s Ä t Ä T by a system .P F ; X; B/ consisting of a complete filtered probability space P F with a filtration F D .F t / 0Ät ÄT satisfying the usual conditions, an d -dimensional F-adapted continuous stochastic process X D .X t / 0Ät ÄT , and an m-dimensional F-Brownian motion B D .B t / 0Ät ÄT defined on P F . Such systems .P F ; X; B/ are said to be weak solutions of SF I.F; G/, SF I.F ; G/, and SF I .F; G/, respectively. If
The Girsanov's theorem is useful as well in the general theory of stochastic analysis as well in its applications. We show here that it can be also applied to the theory of stochastic differential inclusions. In particular, we obtain some special properties of sets of weak solutions to some type of these inclusions.
Necessary and sufficient conditions for the existence of weak solutions to stochastic differential inclusions with convex right-hand sides are given. The main results of the paper deal with the weak compactness with respect to the convergence in distribution of solution sets to such inclusions.
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