Approximation of Euler-Maruyama for one-dimensional stochastic differential equations involving the local times of the unknown process

In this paper, we consider both, the strong and weak convergence of the Euler-Maruyama approximation for one-dimensional stochastic differential equations involving the local times of the unknown process. We use a transformation in order to remove the local time Lat from the stochastic differential equations of type X-t = X-0 + integral(t)(0) phi(X-s)dB(s) +root(R) v(da)L-t(a). Here B is a one-dimensional Brownian motion, phi : R -> R is a bounded measurable function, and v is a bounded measure on R. We provide the approximation of Euler Maruyama for the stochastic differential equations without local time. After that, we conclude the approximation of Euler-Maruyama X-t(n) of the above mentioned equation, and we provide the rate of strong convergence Error = E vertical bar X-T - X-T(n)vertical bar, and the rate of weak convergence Error = E vertical bar G(X-T) - G(X-T(n))vertical bar, for any function G : R -> R of bounded variation.