Reverse Time Diffusions
نویسنده
چکیده
This paper is a continuation of work commenced in [l] and considers a diffusion in R" given as the solution of a family of stochastic differential equations. The problem discussed is to suppose the direction of the time parameter is reversed, that is, time evolves in anegative direction and the same diffusion process is observed, with the filtration generated by the reversed process. There is an extensive literature on time reversal of Markov processes. See [3, 161. The Markov property states that past and future are conditionally independent given the present state, so the reverse time process is again Markov. However, simple examples, (see [16]), show that, for example, the strong Markov property is not preserved by time reversal. Consequently, some basic properties are not preserved, and for diffusions it is not clear that the reverse time process is again a diffusion. This paper gives conditions under which this is so, and in that case derives the reverse time stochastic differential equations giving the reverse time diffusion. Results of this kind have been obtained by, among others, Lindquist and Picci [lo], for the stationary linear case, and by Anderson [I], under the assumptions that the transition density exists and the associated Kolmogorov equations have unique solutions. There is a rather formal derivation, [2], by Castanon, who also assumes the transition densities exist. However, there are gaps in Castanon's work, about
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