A Small Delay and Correlation Time Limit of Stochastic Differential Delay Equations with State-Dependent Colored Noise

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.