On the Solution of Stochastic Functional Differential Equations via Memory Gap

TL;DR

This paper introduces an approximation method with a memory gap to prove the existence of solutions for stochastic functional differential equations, facilitating the extension from discrete delay to continuous memory models.

Contribution

It provides an alternative proof technique using a memory gap approximation scheme for stochastic functional differential equations under Lipschitz conditions.

Findings

Established strong convergence of the approximation scheme

Enabled extension from discrete delay to continuous full memory models

Provided a practical method for solving stochastic functional differential equations

Abstract

We present an alternative proof for the existence of solutions of stochastic functional differential equations satisfying a global Lipschitz condition. The proof is based on an approximation scheme in which the continuous path dependence does not go up to the present: there is a memory gap. Strong convergence is obtained by closing the gap. Such approximation is particularly useful when extending stochastic models with discrete delay to models with continuous full finite memory.