Abstract
It is shown that if a diffusion process, $\{X_t: 0 \leq t \leq 1\}$, on $R^d$ satisfies $dX_t = b(t, X_t) dt + \sigma (t, X_t) dw_t$ then the reversed process, $\{\bar{X}_t: 0 \leq t \leq 1\}$ where $\bar{X}_t = X_{1 - t}$, is again a diffusion with drift $\bar{b}$ and diffusion coefficient $\bar{\sigma}$, provided some mild conditions on $b, \sigma$, and $p_0$, the density of the law of $X_0$, hold. Moreover $\bar{b}$ and $\bar\sigma$ are identified.
Citation
U. G. Haussmann. E. Pardoux. "Time Reversal of Diffusions." Ann. Probab. 14 (4) 1188 - 1205, October, 1986. https://doi.org/10.1214/aop/1176992362
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