Abstract
Given a forward ( = usual) stochastic differential equation (SDE), we consider, in this paper, an associated backward SDE. Let E;s,t(x),t∈[s, ∞) be the solution of an SDE on a manifold M: with the initial condition ξs,s(x) =x. Here X 0,…,X r are smooth vector fields, (B t 1,…,B t 1) is a standard r-dimensional Brownian motion and o denotes the Stratonovich integral. We show that the solution E;s,t satisfies the backward SDE: where ξst the differential of the map Es,t(·)M→M and [dcirc]B s j denotes the backward stochastic integral. The result is applied to getting a necessary and sufficient condition that the map ξs,t: defines a diffeomorphism of M a.s.
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