Euler-Maruyama Approximation for Diffusion Process Generated by Divergence form Operator with Discontinuous Coefficients

Abstract

Weconsiderthe Euler-Maruyamaapproximationfor time-inhomogeneous one-dimensional stochastic differential equations involving the local time (SDELT), generated by divergence form operators with discontinuous coefficients at zero. We use a space transform in order to remove the local time L⁰_t from the stochastic differential equation of type dX_t = σ(t,X_t)dB_t + 1/2σ(t,X_t)σ′_x(t,X_t)dt + β(t)dL⁰_t(X). After that we use a transformation technique that removes the discontinuity from the drift of the new auxiliary equation without local time, such that the coefficients of the obtained time-inhomogeneous SDE are Lipschitz continuous in space. Thus the Euler-Maruyama method can be applied and we provide the rate of strong convergence desired.

Key words and phrases. stochastic differential equations, local time, Euler-Maruyama method, discontinuous coefficients.

2010 Mathematics Subject Classification. 60H35, 60H10, 60J55, 65C30