Optimal Strong Approximation of the One-dimensional Squared {B}essel Process

TL;DR

This paper investigates the strong approximation of a one-dimensional squared Bessel process, revealing that adaptive algorithms significantly outperform non-adaptive ones in convergence rate, with implications for CIR process parameter effects.

Contribution

It demonstrates that adaptive numerical methods achieve infinitely fast convergence for the squared Bessel process, unlike equidistant grid methods, and analyzes parameter influence on convergence.

Findings

Adaptive algorithms have infinite convergence rate.

Equidistant grid algorithms have convergence rate of 1/2.

Process parameters influence the approximation convergence rate.

Abstract

We consider the one-dimensional squared Bessel process given by the stochastic differential equation (SDE) \begin{align*} dX_t = 1\,dt + 2\sqrt{X_t}\,dW_t, \quad X_0=x_0, \quad t\in[0,1], \end{align*} and study strong (pathwise) approximation of the solution $X$ at the final time point $t=1$. This SDE is a particular instance of a Cox-Ingersoll-Ross (CIR) process where the boundary point zero is accessible. We consider numerical methods that have access to values of the driving Brownian motion $W$ at a finite number of time points. We show that the polynomial convergence rate of the $n$-th minimal errors for the class of adaptive algorithms as well as for the class of algorithms that rely on equidistant grids are equal to infinity and $1/2$, respectively. This shows that adaption results in a tremendously improved convergence rate. As a by-product, we obtain that the parameters appearing in the CIR process affect the convergence rate of strong approximation.