Time reversal of diffusions. (English) Zbl 0607.60065

Let \(\{X_ t: 0\leq t\leq 1\}\) be a diffusion process on \({\mathbb{R}}^ d\) satisfying \(dX_ t=b(t,X_ t)dt+\sigma (t,X_ t)dW_ t\), where \(\{W_ t: 0\leq t\leq 1\}\) is a standard Brownian motion, b is the drift and \(\sigma\) diffusion coefficient. Put \(\bar X_ t=X_{1-t}\), the reversed process. The authors give sufficient but mild conditions on b, \(\sigma\) and \(p_ 0\), the density of the initial distribution of \(X_ 0\), to prove that \(\bar X\) is again a diffusion. That is, there exist \(\bar b,\) \({\bar \sigma}\) and a Brownian motion \(\bar W\) such that \(d\bar X_ t=\bar b(t,X_ t)dt+{\bar \sigma}(t,\bar X_ t)d\bar W_ t\). Moreover, \(\bar b\) and \({\bar \sigma}\) are also identified.
Reviewer: Lou Jiann Hua


60J60 Diffusion processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
35K15 Initial value problems for second-order parabolic equations
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