A Note on Explicit Milstein-Type Scheme for Stochastic Differential Equation with Markovian Switching

TL;DR

This paper introduces an explicit Milstein-type numerical scheme for stochastic differential equations with Markovian switching, proving its strong convergence rate of 1.0 under mild regularity conditions without relying on Itô-Taylor expansion.

Contribution

The paper develops a new explicit Milstein-type scheme for SDEs with Markovian switching and proves its strong convergence rate under relaxed coefficient regularity assumptions.

Findings

Strong convergence rate of 1.0 established

Scheme derived without Itô-Taylor expansion

Applicable under mild regularity conditions

Abstract

An explicit Milstein-type scheme for stochastic differential equation with Markovian switching is derived and its strong convergence in $\mathcal{L}^2$-sense is established without using It\^o-Taylor expansion formula. Rate of strong convergence is shown to be equal to $1.0$ under the assumptions that coefficients satisfy mild regularity conditions. More precisely, coefficients are assumed to be only once differentiable which are more relaxed conditions than those made in existing literature.