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Approximate solution methods for linear stochastic difference equations

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  • Roerdink, J.B.T.M.

Abstract

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 ατc, where τc 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.

Suggested Citation

  • Roerdink, J.B.T.M., 1983. "Approximate solution methods for linear stochastic difference equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 119(3), pages 455-484.
    • Handle: RePEc:eee:phsmap:v:119:y:1983:i:3:p:455-484
    DOI: 10.1016/0378-4371(83)90103-6
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