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The coefficients of the stochastic differential equations with Markovianswitching (SDEwMS) additionally depend on a Markov chain and there is no notionof differentiating such functions with respect to the Markov chain. Inparticular, this implies that the It\^o-Taylor expansion for SDEwMS is not astraightforward extension of the It\^o-Taylor expansion for stochasticdifferential equations (SDEs). Further, higher-order numerical schemes forSDEwMS are not available in the literature, perhaps because of the absence ofthe It\^o-Taylor expansion. In this article, first, we overcome thesechallenges and derive the It\^o-Taylor expansion for SDEwMS, under somesuitable regularity assumptions on the coefficients, by developing newtechniques. Secondly, we demonstrate an application of our first result on theIt\^o-Taylor expansion in the numerical approximations of SDEwMS. We derive anexplicit scheme for SDEwMS using the It\^o-Taylor expansion and show that thestrong rate of convergence of our scheme is equal to$\gamma\in\{n/2:n\in\mathbb{N}\}$ under some suitable Lipschitz-type conditionson the coefficients and their derivatives. It is worth mentioning thatdesigning and analysis of the It\^o-Taylor expansion and the$\gamma\in\{n/2:n\in\mathbb{N}\}$-order scheme for SDEwMS become much morecomplex and involved due to the entangling of continuous dynamics and discreteevents. Finally, our results coincide with the corresponding results on SDEswhen the state of the Markov chain is a singleton set.

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