Multiplicative Stochastic Processes Involving the Time-Derivative of a Markov Process.

摘要

The characteristic functional of the derivative phi(t) of a Markov process phi(t) and the related multiplicative process sigma(t), which obeys the stochastic differential equation sigma(t) = (A + phi(t)B)sigma(t), have been studied. Exact equations for the marginal characteristic functional and the marginal average of sigma(t) are derived. The first equation is applied to obtain a set of equations for the marginal moments of phi(t) in terms of the prescribed properties of phi(t). It is illustrated by an example how these equations can be solved, and it is shown in general that phi(t) is delta-correlated, with a smooth background. The equation of motion for the marginal average of sigma(t) is solved for various cases, and it is shown how closed-form analytical expressions for the average can be obtained. (Author)