A small delay and correlation time limit of stochastic differential delay equations with state-dependent colored noise

TL;DR

This paper analyzes the limiting behavior of stochastic differential delay equations with state-dependent colored noise as delays and correlation times approach zero, motivated by electrical circuit experiments.

Contribution

It derives the limit of SDDEs with colored noise and delays, extending existing theory to more realistic noise models with state dependence.

Findings

Limit established for SDDEs with colored noise and delays

Uses Kurtz-Protter convergence theorem and Peligrad-Utev inequality

Provides a rigorous mathematical framework for noisy, delayed feedback systems

Abstract

We consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. An Ornstein-Uhlenbeck process is used to model the colored noise. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.