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An SDE approximation for stochastic differential delay equations with state-dependent colored noise. (English) Zbl 1356.60091

Summary: 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 T. G. Kurtz and P. Protter [Ann. Probab. 19, No. 3, 1035–1070 (1991; Zbl 0742.60053)]. 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.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness

Citations:

Zbl 0742.60053
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