On It\^o-Taylor expansion for stochastic differential equations with Markovian switching and its application in $\gamma\in\{n/2:n \in\mathbb{N}\}$-order scheme

TL;DR

This paper develops an Itô-Taylor expansion for stochastic differential equations with Markovian switching, enabling higher-order numerical schemes with proven convergence rates, addressing a gap in the literature caused by the complexity of combined continuous and discrete dynamics.

Contribution

The paper introduces the first Itô-Taylor expansion for SDEs with Markovian switching and constructs explicit high-order schemes with proven convergence rates.

Findings

Derived the Itô-Taylor expansion for SDEwMS under regularity conditions.

Constructed an explicit numerical scheme with strong convergence rate γ.

Results reduce to classical SDE cases when the Markov chain is trivial.

Abstract

The coefficients of the stochastic differential equations with Markovian switching (SDEwMS) additionally depend on a Markov chain and there is no notion of differentiating such functions with respect to the Markov chain. In particular, this implies that the It\^o-Taylor expansion for SDEwMS is not a straightforward extension of the It\^o-Taylor expansion for stochastic differential equations (SDEs). Further, higher-order numerical schemes for SDEwMS are not available in the literature, perhaps because of the absence of the It\^o-Taylor expansion. In this article, first, we overcome these challenges and derive the It\^o-Taylor expansion for SDEwMS, under some suitable regularity assumptions on the coefficients, by developing new techniques. Secondly, we demonstrate an application of our first result on the It\^o-Taylor expansion in the numerical approximations of SDEwMS. We derive an explicit scheme for SDEwMS using the It\^o-Taylor expansion and show that the strong rate of convergence of our scheme is equal to $\gamma\in\{n/2:n\in\mathbb{N}\}$ under some suitable Lipschitz-type conditions on the coefficients and their derivatives. It is worth mentioning that designing and analysis of the It\^o-Taylor expansion and the $\gamma\in\{n/2:n\in\mathbb{N}\}$-order scheme for SDEwMS become much more complex and involved due to the entangling of continuous dynamics and discrete events. Finally, our results coincide with the corresponding results on SDEs when the state of the Markov chain is a singleton set.