An SDE Approximation for Stochastic Differential Delay Equations with State-Dependent Colored Noise

O. Duman, A. McDaniel, G. Volpe, J. Wehr

2016, v.22, Issue 3, 595-628

ABSTRACT

We consider a general multidimensional stochastic differential delay equation (SDDE) with state-dependent colored noises. We approximate it by a stochastic differential equation (SDE) system and calculate its limit as the time delays and the correlation times of the noises go to zero. The main result is proven using a theorem about convergence of stochastic integrals by Kurtz and Protter. It formalizes and extends a result that has been obtained in the analysis of a noisy electrical circuit with delayed state-dependent noise, and may be used as a working SDE approximation of an SDDE modeling a real system where noises are correlated in time and whose response to noise sources depends on the system's state at a previous time.

Keywords: stochastic differential equations, stochastic differential delay equations, colored noise, noise-induced drift

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