Approximation of Stochastic Delay Differential Systems by a Stochastic System Without Delay

Abstract

In this paper, we propose a scheme for approximating the solutions of stochastic differential equations with delay by solutions of stochastic differential equations without delay. Stochastic delay differential equations play a crucial role in modeling real-world processes where the evolution depends on past states, introducing complexities due to their infinite-dimensional phase space. To overcome these difficulties, we develop an approach based on approximating the delay system by an ordinary differential equation system of increased dimension. Our main result is to prove that, under certain conditions, the solutions of the approximating system converge in the mean square sense to the solutions of the original delay system. This approach allows for effective analysis and modeling of stochastic systems with delay using finite-dimensional stochastic differential equations without delay.