Reverse - Time Diffusion Equation Models *

Stochastic differential equations have a built-in direction of time flow since future increments in the driving process are assumed independent of present and past values of the process defined by the solution of the equation. The differential equations are thought of as evolving forward in time, normally from some fixed initial time, and the integral representation of a solution, involving as it does an Ito integral, emphasizes again, via the detailed approximation rule for the integral, the forward time flow. In this paper, we discuss reverse-time stochastic differential equations, and for a wide variety of diffusion processes, we show that each (forwardtime) representation of a diffusion process generates a reverse-time representation as well. The only sorts of restrictions needed are those which ensure that the Kolmogorov equations for associated probability densities (not just distribution functions) all have unique smooth solutions; such restrictions, though hard to translate into requirements on the diffusion and drift quantities, seem nevertheless intrinsic. Results in this vein for diffusion processes described by linear stochastic differential equations have been recently developed, see especially [I] but also [2-91. Some of these references contain applications of the reverse-time models to problems of