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On backward stochastic differential equations

 

作者: Hiroshi Kunitha,  

 

期刊: Stochastics  (Taylor Available online 1982)
卷期: Volume 6, issue 3-4  

页码: 293-313

 

ISSN:0090-9491

 

年代: 1982

 

DOI:10.1080/17442508208833209

 

出版商: Gordon and Breach Science Publishers Inc

 

数据来源: Taylor

 

摘要:

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. HereX0,…,Xrare smooth vector fields, (Bt1,…,Bt1) is a standard r-dimensional Brownian motion and o denotes the Stratonovich integral. We show that the solution E;s,tsatisfies the backward SDE:where ξstthe differential of the map Es,t(·)M→Mand [dcirc]Bsjdenotes the backward stochastic integral. The result is applied to getting a necessary and sufficient condition that the map ξs,t: defines a diffeomorphism ofMa.s.

 

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