On Explicit Tamed Milstein-type scheme for Stochastic Differential Equation with Markovian Switching

TL;DR

This paper introduces a new tamed Milstein-type numerical scheme for stochastic differential equations with Markovian switching, achieving a strong convergence rate of 1.0 even with super-linear drift growth.

Contribution

It develops a novel tamed Milstein scheme that handles jumps and reduced regularity, providing optimal convergence for SDEs with Markovian switching.

Findings

Strong convergence rate of 1.0 established

Handles super-linear drift growth effectively

Develops techniques for jump and regularity challenges

Abstract

We propose a new tamed Milstein-type scheme for stochastic differential equation with Markovian switching when drift coefficient is assumed to grow super-linearly. The strong rate of convergence is shown to be equal to $1.0$ under mild regularity (e.g. once differentiability) requirements on drift and diffusion coefficients. Novel techniques are developed to tackle two-fold difficulties arising due to jumps of the Markov chain and the reduction of regularity requirements on the coefficients.