Strong convergence of the adaptive Milstein method for nonlinear stochastic differential equations with piecewise continuous arguments

TL;DR

This paper introduces an adaptive Milstein method for nonlinear stochastic differential equations with piecewise continuous arguments, proving its strong convergence and optimal rate, and demonstrating its effectiveness through numerical experiments.

Contribution

It is the first to develop a variable step size numerical method for nonlinear SDEPCAs with superlinear growth, ensuring strong convergence and optimal rate.

Findings

The adaptive Milstein method is strongly convergent in $L_p$ for $p \\ge 2$.

The convergence rate matches that of the explicit Milstein scheme with Lipschitz coefficients.

Numerical experiments confirm the theoretical convergence and effectiveness of the method.

Abstract

In this work, an adaptive time-stepping Milstein method is constructed for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift is one-sided Lipschitz continuous and the diffusion does not impose the commutativity condition. It is widely recognized that explicit Euler or Milstein methods may blow up when the system exhibits superlinear growth, and modifications are needed. Hence we propose an adaptive variant to deal with the case of superlinear growth drift coefficient. To the best of our knowledge, this is the first work to develop a numerical method with variable step sizes for nonlinear SDEPCAs. It is proven that the adaptive Milstein method is strongly convergent in the sense of $L_p, p\ge 2$, and the convergence rate is optimal, which is consistent with the order of the explicit Milstein scheme with globally Lipschitz coefficients. Finally, several numerical experiments are presented to support the theoretical analysis.