Adaptive Mesh Construction for the Numerical Solution of Stochastic Differential Equations with Markovian Switching

TL;DR

This paper introduces an adaptive mesh method for numerically solving nonlinear stochastic differential equations with Markovian switching, improving accuracy by locally adjusting timesteps to rapid changes and switching events.

Contribution

It proposes a hybrid explicit-implicit scheme with adaptive meshes that incorporate Markov switching times, achieving strong mean-square convergence of order 1.

Findings

Successfully constructed an order 1 numerical method.

Applied the method to model telomere length dynamics.

Demonstrated improved accuracy with adaptive meshing.

Abstract

We demonstrate an approach to the numerical solution of nonlinear stochastic differential equations with Markovian switching. Such equations describe the stochastic dynamics of processes where the drift and diffusion coefficients are subject to random state changes according to a Markov chain with finite state space. We propose a variant of the Jump Adapted-Adaptive approach introduced by K, Lord, \& Sun~(2025) to construct nonuniform meshes for explicit numerical schemes that adjust timesteps locally to rapid changes in the numerical solution and which also incorporate the switching times of an underlying Markov chain as meshpoints. It is shown that a hybrid scheme using such a mesh that combines an efficient explicit method (to be used frequently) and a potentially inefficient backstop method (to be used occasionally) will display strong convergence in mean-square of order $\delta$ if both methods satisfy a mean-square consistency condition of the same order in the absence of switching. We demonstrate the construction of an order $\delta=1$ method of this type and apply it to generate empirical distributions of a nonlinear SDE model of telomere length in DNA replication.