Milstein scheme for stochastic differential equation with Markovian switching and Lévy noise

First, we establish the Ito's formula for stochastic differential equations with Markovian switching and Levy noise which is then used to derive a first-order scheme (Milstein scheme). The moment stability of the Milstein scheme is investigated and its rate of convergence is proved to be 1.0 when the coefficients and their derivatives are Lipschitz continuous. The intertwining of the cadlag (solution) process with the discontinuous dynamics of the underlying Markov chain gives rise to several challenges and new techniques are developed to deal with them. We also provide a discussion on the practical implementation of the proposed Milstein scheme under the diffusion and jump commutativity conditions. (c) 2024 Elsevier Inc. All rights reserved.