A Convergence result for the Euler-Maruyama method of one-dimensional stochastic differential equations involving the local time of the unknown process at zero

In this paper, we study the strong rate of convergence for the Euler-Maruyama approximation of a class of one-dimensional stochastic differential equations involving the local time (SDELT) of the unknown process, corresponding to divergence form operator with a discontinuous coefficients at zero. We use a space transform in order to remove the local time L_t⁰ from the stochastic differential equation of type, dX_t = σ(X_t)dB_t + ψ(X_t)d_t + βdL_t⁰ (X). Here B is a standard one-dimensional Brownian motion. σ and ψ are a bounded measurable functions, and β ∈ (-1, 1) . We provide the approximation of Euler-Maruyama for the stochastic differential equation without local time. After that the approximation can be transformed back, giving an approximation of Euler-Maruyama Xnt to the solution of the original SDELT, and we provide the rate of strong convergence Error = E[|X_Tⁿ -X_T|].