Some additional remarks on the cumulant expansion for linear stochastic differential equations

Abstract

We summarize our previous results on cumular expasions for linear stochastic differential equations with correlated multipliclative and additive noise. The application of the general formulas to equations with statistically independent multiplicative and additive noise is reconsidered in detail, because our earlier results for this case are only valid in special situations. Moreover, a second partially time ordered cumulant is introduced, which is appropriate for the calculation of multi-time averages of the solution process and which differs from the one to be used for single-time averages. By the two supplements just mentioned we complete our earlier investigations on the subject.