Approximate solution methods for linear stochastic difference equations: I. Moments

The cumulant expansion for linear stochastic differential equations is extended to the case of linear stochastic difference equations. We consider a vector difference equation, which contains a deterministic matrix A₀ and a random perturbation matrix A₁(t). The expansion proceeds in powers of ατ_c, where τ_c is the correlation time of the fluctuations in A₁(t) and α a measure for their strength. Compared to the differential case, additional cumulants occur in the expansion. Moreover one has to distinguish between a nonsingular and a singular A₀. We also discuss a limiting situation in which the stochastic difference equation can be replaced by a stochastic differential equation. The derivation is not restricted to the case where in the limit the stochastic parameters in the difference equation are replaced by white noise.