Convergence of the tamed Euler scheme for stochastic differential equations with Piecewise Continuous Arguments under non-Lipschitz continuous coefficients

TL;DR

This paper proves that the tamed Euler method converges strongly with order one half when applied to stochastic differential equations with piecewise continuous arguments and superlinear coefficients, extending previous results to a broader class of SDEs.

Contribution

It extends the convergence analysis of the tamed Euler method to SEPCAs with superlinear coefficients, demonstrating strong convergence with order one half.

Findings

Tamed Euler method converges strongly for SEPCAs with superlinear coefficients.

Convergence order of the method is one half.

Applicable to equations with piecewise continuous arguments.

Abstract

Recently, Martin Hutzenthaler pointed out that the explicit Euler method fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with superlinearly growing and globally one sided Lipschitz drift coefficient. Afterwards, he proposed an explicit and easily implementable Euler method, i.e tamed Euler method, for such an SDE and showed that this method converges strongly with order of one half. In this paper, we use the tamed Euler method to solve the stochastic differential equations with piecewise continuous arguments (SEPCAs) with superlinearly growing coefficients and prove that this method is convergent with strong order one half.