The characteristic functional of the derivative \({\dot \phi}\)(t) of a Markov process \(\phi\) (t) and the related multiplicative process \(\sigma\) (t), which obeys the stochastic differential equation i\({\dot \sigma}\)(t)\(=(A+{\dot \phi}(t)B)\sigma (t)\), 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 \({\dot \phi}\)(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 \({\dot \phi}\)(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 \(<\sigma (t)>\) can be obtained.