Stochastic differential inclusions in terms of infinitesimal generators and mean derivatives

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 th...

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Bibliographic Details
Published in:Applicable analysis 2009-01, Vol.88 (1), p.89-105
Main Authors: Azarina, Svetlana V., Gliklikh, Yuri E.
Format: Article
Language:English
Subjects:
infinitesimal generators
mean derivatives
stochastic differential inclusions
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Summary: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.
ISSN:0003-6811
1563-504X
DOI:10.1080/00036810802556795