Convergence of solutions of mixed stochastic delay differential equations with applications

Abstract

The paper is concerned with a mixed stochastic delay differential equation involving both a Wiener process and a $\gamma$-Hölder 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.