Optimal strong approximation of the one-dimensional squared Bessel process

We consider the one-dimensional squared Bessel process given by the stochastic differential equation (SDE)\[{\mathrm{d}X}_t = 1\mathrm{d}t + 2 \sqrt{X_t} {\mathrm{d}W}_t \textrm{,} \qquad X_0 = x_0 \textrm{,} \qquad t \in [0,1] \textrm{,}\]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.

Keywords

Cox–Ingersoll–Ross process, strong approximation, $n$-th minimal error, adaptive algorithm, reflected Brownian motion

2010 Mathematics Subject Classification

60H10, 65C30

Full Text (PDF format)

Received 21 June 2016

Accepted 30 May 2017

Published 20 December 2017