Convergence rate of Euler scheme for time-inhomogeneous SDEs involving the local time of the unknown process

In this paper, we are concerned with strong convergence rate of Euler scheme for time-inhomogeneous one-dimensional stochastic differential equations involving the local time (SDELT) of the unknown process at point zero. We use a space transform in order to remove the local time from this class of stochastic differential equations. We provide the approximation of Euler for the stochastic differential equation without local time. After that the approximation can be transformed back, giving an approximation of Euler to the solution of the original SDELT, and we provide the rate of strong convergence.