A New Discretization Scheme for One Dimensional Stochastic Differential Equations Using Time Change Method

TL;DR

This paper introduces a novel numerical discretization scheme for one-dimensional SDEs based on a time change method, achieving strong convergence even when the diffusion coefficient is only Hölder continuous with exponent less than 1/2.

Contribution

It presents the first scheme to attain strong convergence for SDEs with Hölder continuous coefficients where the exponent is below 1/2, expanding numerical options for such equations.

Findings

Achieves strong convergence for coefficients with Hölder exponent less than 1/2.

Provides convergence rates for bounded, Hölder continuous diffusion coefficients.

Enables approximation of weak solutions without requiring strong solutions.

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 a SDE with a time changed Brownian motion, dated back to Doeblin (1940). In cases where the diffusion coefficient is bounded and $\beta$-H\"{o}lder continuous with $0 < \beta \leq 1$, we provide the rate of strong convergence. An advantage of our approach is that we approximate the weak solution, which enables us to treat a SDE with no strong solution. Our scheme is the first to achieve the strong convergence for the case $0 < \beta < 1/2$.