A note on Euler approximations for stochastic differential equations with delay

TL;DR

This paper establishes existence, uniqueness, and convergence properties of Euler approximations for stochastic delay differential equations, including convergence rates under various conditions.

Contribution

It provides the first comprehensive proof of convergence and rates for Euler schemes applied to stochastic delay differential equations under broad conditions.

Findings

Existence and uniqueness theorem for stochastic delay differential equations.

Proof of convergence of Euler approximations under general conditions.

Determination of almost sure convergence rates under Lipschitz and monotonicity assumptions.

Abstract

An existence and uniqueness theorem for a class of stochastic delay differential equations is presented, and the convergence of Euler approximations for these equations is proved under general conditions. Moreover, the rate of almost sure convergence is obtained under local Lipschitz and also under monotonicity conditions.