Time reversal of diffusions. (English) Zbl 0607.60065

Haussmann, U. G.; Pardoux, E.

Ann. Probab. 14, 1188-1205 (1986).

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

MSC:

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

Keywords:

time reversal; martingale problem; Kolmogorov equation