Projected Euler method for stochastic delay differential equation under a global monotonicity condition

TL;DR

This paper extends the projected Euler-Maruyama method to stochastic delay differential equations with nonlinear coefficients under a global monotonicity condition, proving convergence and demonstrating effectiveness through numerical examples.

Contribution

It generalizes C-stability and B-consistency concepts to delay equations and establishes a convergence order of 1/2 for the method.

Findings

Method is convergent with order 1/2.

Numerical examples confirm theoretical results.

Applicable to equations with highly nonlinear coefficients.

Abstract

This paper investigates projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition. This condition admits some equations with highly nonlinear drift and diffusion coefficients. We appropriately generalized the idea of C-stability and B-consistency given by Beyn et al. [J. Sci. Comput. 67 (2016), no. 3, 955-987] to the case with delay. Moreover, the method is proved to be convergent with order $\frac{1}{2}$ in a succinct way. Finally, some numerical examples are included to illustrate the obtained theoretical results.