Stochastic Differential Equations (SDEs) have become standard models for financial quantities such as stock prices, interest rates, and their derivatives. However, their analytical solutions are very rare, and it is possible to simulate its trajectories through a discretization scheme. Hence, both in literature and practice, most attention has been directed to discrete time approximations of Stochastic Differential Equations (SDEs). In such numerical treatment of stochastic differential equations, it is of theoretical and practical importance to estimate the rate of convergence of the discrete-time approximation.

The rate of convergence of the Euler scheme has been studied for various convergence criteria. Talay and Tubaro showed convergence rate of the expectation of functionals of solutions of SDEs with smooth coefficients. Bally and Talay studied convergence rate of the distribution function, density and the error analysis. And, the detailed review was provided by Kloeden and Platen. Protter and Talay studied the Euler scheme for SDE driven by Levy processes. Kubilius and Platen estimated the speed of convergence of the Euler approximation for diffusion processes with jump component which have Holder continuous coefficients, and there are other plenty of research papers and detailed reviews in this field.

However, the case of SDEs with discontinuous coefficients has hardly been investigated. Chan and Stramer studied the weak convergence of the Euler scheme for SDEs with coefficients satisfying some regularity conditions. Recently, Dai Taguchi investigated stability problem for one-dimensional SDEs with discontinuous drift. Hoang-Long and Dai Taguchi studied Euler scheme for SDEs with very irregular drift and constant diffusion coefficients. And, Gunther Leobacher and Michaela Szolgyenyi studied the convergence of the Euler scheme for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient.