Optimal Strong Approximation of the One-dimensional Squared {B}essel Process
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 at the final time point . 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 at a finite number of time points. We show that the polynomial convergence rate of the -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 , 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.
Cite
@article{arxiv.1601.01455,
title = {Optimal Strong Approximation of the One-dimensional Squared {B}essel Process},
author = {Mario Hefter and André Herzwurm},
journal= {arXiv preprint arXiv:1601.01455},
year = {2016}
}