Convergence of solutions of mixed stochastic delay differential equations with applications

TL;DR

This paper studies mixed stochastic delay differential equations driven by Wiener and fractional Brownian motions, proving solution stability, convergence of delayed solutions to non-delayed ones, and Euler approximation convergence.

Contribution

It introduces new stability results for solutions and demonstrates convergence of solutions and numerical schemes in mixed stochastic delay differential equations.

Findings

Solutions depend continuously on coefficients and initial data.

Solutions with vanishing delay converge to delay-free solutions.

Euler approximations converge to true solutions.

Abstract

The paper is concerned with a mixed stochastic delay differential equation involving both a Wiener process and a $\gamma$-H\"older continuous process with $\gamma>1/2$ (e.g. a fractional Brownian motion with Hurst parameter greater than $1/2$). It is shown that its solution depends continuously on the coefficients and the initial data. Two applications of this result are given: the convergence of solutions to equations with vanishing delay to the solution of equation without delay and the convergence of Euler approximations for mixed stochastic differential equations. As a side result of independent interest, the integrability of solution to mixed stochastic delay differential equations is established.