ITÔ-TAYLOR EXPANSION AND EXPLICIT NUMERICAL SCHEMES FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH MARKOVIAN SWITCHING

Abstract

Stochastic di erential equations with Markovian switching (SDEwMS) exhibit both the continuous dynamics, related to the state of the process and the discrete dynamics, related to the state of the Markov chain which is present in the coe cients of such equations. Over the past few decades, such regime switching models have found applications in a wide variety of real life situations encountered in automatic control, manufacturing systems, nancial engineering, economics, mathematical biology, etc. The entangling of the continuous and discrete dynamics in SDEwMS brings additional complexities in their analysis, particularly when one aims to construct numerical approximations of order higher than 1/2, mainly due to the inability to di erentiate Markov chain dependent functions. In part one, we derive an explicit Milstein scheme for SDEwMS without using the Itô's formula for SDEwMS and show that its strong rate of convergence in L2-sense is equal to 1.0. The drift and di usion coe cients along with their derivatives are assumed to be Lipschitz continuous which is more relaxed than the corresponding results available in the literature. Further, we extend this result to the case when the drift coe cient and its derivatives satisfy polynomial Lipschitz conditions by proposing a tamed Milstein scheme for SDEwMS. In part two, we provide the Itô -Taylor expansion for SDEwMS by repeatedly using the Itô's formula on the coe cients. However, this is not a straightforward extension of the well-known result on the Itô -Taylor expansion for SDEs, not only due to the absence of the notion of di erentiability of Markov chain dependent functions but also the appearance of a new integral with respect to an optional process associated with the switching of the Markov chain. In addition, the state(s) of the chain start appearing in the integrands and the integrators which further complicates our analysis. To deal with these challenges, we develop new tools and techniques and give the derivation and proof of the Itô-Taylor expansion for SDEwMS under some suitable regularity assumptions on the coe cients. As an application of the Itô -Taylor expansion, we derive an explicit numerical scheme of order γ ∈ {ξ/2 : ξ ∈ N} for SDEwMS and investigate its strong convergence in L2-sense under some suitable Lipschitz conditions on the coe cients and their derivatives. These ndings are consistent with results on SDEs when the state space of the chain is a singleton set.