A note on the Euler–Maruyama scheme for 1-dimensional stochastic differential equations involving the local time of the unknown process

Abstract

In this paper, we study both strong and weak rate of convergence of Euler-Maruyama scheme for one-dimensional stochastic differential equations involving the local time (SDELT) of the unknown process, whose the solution corresponds to divergence form operator with a discontinuous coefficients at point ξ ∈ ℝ. The paper is concerned with stochastic differential equation with local time of type, where is the local time of X in {ξ}. The main idea of this paper is to use a space transform in order to transform the original SDELT to a new auxiliary equation with discontinuous coefficients but without local time. Then, we apply the Euler-Maruyama scheme to this new equation. Finally, we obtain an approximation of the original SDELT by transforming the approximation of the auxiliary equation.