Approximation of Euler Maruyama for one-dimensional stochastic differential equations involving the maximum process

The aim of this paper is to show the approximation of Euler-Maruyama X-t(n) for one-dimensional stochastic differential equations involving the maximum process. In addition to that it proves the strong convergence of the Euler-Maruyama whose both drift and diffusion coefficients are Lipschitz. After that, it generalizes to the non-Lipschitz case.