Strong approximation of particular one-dimensional diffusions

TL;DR

This paper introduces a new, explicit method for strongly approximating one-dimensional diffusions, including Brownian motion and related SDEs, with proven convergence and practical implementation details.

Contribution

It develops a novel, easy-to-implement technique for path approximation of 1D diffusions, controlling step count and convergence, combining exit time analysis and renewal theory.

Findings

The scheme effectively approximates Brownian motion and related diffusions.

Convergence theorems guarantee accuracy of the approximation.

Numerical examples demonstrate practical applicability.

Abstract

This paper develops a new technique for the path approximation of one-dimensional stochastic processes, more precisely the Brownian motion and families of stochastic differential equations sharply linked to the Brownian motion (usually known as L and G-classes). We are interested here in the $\epsilon$-strong approximation. We propose an explicit and easy to implement procedure that constructs jointly, the sequences of exit times and corresponding exit positions of some well chosen domains. The main results control the number of steps to cover a fixed time interval and the convergence theorems for our scheme. We combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also described in order to complete the theoretical results.