A new discretization scheme for one dimensional stochastic differential equations using time change method

Abstract

We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of an SDE using a time-changed Brownian motion, which dates back to Doeblin (1940). In cases where the diffusion coefficient is bounded and is β-Hölder continuous with $0<\mathit{\beta }\le 1$, we provide the rate of strong convergence. An advantage of our approach is that we approximate the weak solution, which enables us to treat SDEs with no strong solution. Our scheme is the first to achieve strong convergence for the case of $0<\mathit{\beta }<1∕2$.