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

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.