An SDE approximation for stochastic differential delay equations with state-dependent colored noise

TL;DR

This paper develops an SDE approximation for complex stochastic delay equations with state-dependent colored noise, providing a rigorous limit analysis as delays and noise correlations vanish, useful for modeling real systems with correlated noise.

Contribution

It extends existing results by formalizing an SDE approximation for SDDEs with state-dependent colored noise and analyzing its convergence as delays and correlations tend to zero.

Findings

Proves convergence of the SDE approximation using Kurtz and Protter's theorem.

Provides a practical method for approximating SDDEs with correlated noise.

Extends analysis to multidimensional systems with state-dependent delays.

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.