Strong approximation of Bessel processes

TL;DR

This paper introduces a new, efficient algorithm for the path approximation of Bessel processes, extending previous work on Brownian motion, with proven convergence and practical numerical implementations across different dimensions.

Contribution

The authors develop a novel $ ext{ε}$-strong approximation algorithm for Bessel processes that is easy to implement and applicable in any dimension, with separate techniques for integer and non-integer cases.

Findings

The algorithm converges for both integer and non-integer dimensions.

The method provides control over efficiency relative to the parameter ε.

Numerical experiments demonstrate the practical effectiveness of the approach.

Abstract

We consider the path approximation of Bessel processes and develop a new and efficient algorithm. This study is based on a recent work by the authors, on the path approximation of the Brownian motion, and on the construction of specific own techniques. It is part of the family of the so-called $\varepsilon$-strong approximations. More precisely, our approach constructs jointly the sequences of exit times and corresponding exit positions of some well-chosen domains, the construction of these domains being an important step. Based on this procedure, we emphasize an algorithm which is easy to implement. Moreover, we can develop the method for any dimension. We treat separately the integer dimension case and the non integer framework, each situation requiring appropriate techniques. In particular, for both situations, we show the convergence of the scheme and provide the control of the efficiency with respect to the small parameter $\varepsilon$. We expand the theoretical part by a series of numerical developments.