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

Masaaki Fukasawa,Mitsumasa Ikeda

doi:10.1214/21-ecp420

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-Holder continuous with 0<β≤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<β<1∕2.

Highlights

In this article, we provide a numerical method for approximating a weak solution of a one dimensional stochastic differential equation
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)

Summary

Introduction

We provide a numerical method for approximating a weak solution of a one dimensional stochastic differential equation. This work of Doeblin from 1940 was only made public in 2000, the idea was rediscovered and extended in stochastic calculus, and was already in a textbook [10] by Ikeda and Watanabe in 1984, where it was shown that a certain class of one-dimensional SDE of the form (1.1) has a unique solution represented by a time changed Brownian motion, where the time change is given as the solution of a random ordinary differential equation, as we discuss in more detail We use this representation to construct a new approximation scheme for one-dimensional SDEs. For the time homogeneous case, namley, σ(t, x) = σ(x) in (1.1), Engelbert and Schmidt [4] gave an an equivalent condition for weak existence and uniqueness in the sense of probability law, under which the weak solution is represented by a time-changed Brownian motion.

Discretization with time change

Rate of convergence

Findings

Numerical example