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

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 \(t=1\). Th...

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Bibliographic Details
Published inarXiv.org
Main Authors Hefter, Mario, Herzwurm, André
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 07.01.2016
Subjects
Adaptive algorithms
Algorithms
Approximation
Brownian motion
Convergence
Differential equations
Numerical methods
Polynomials
Stochastic processes
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Summary: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.
ISSN:2331-8422