Direct Inversion for the Squared Bessel Process and Applications

TL;DR

This paper introduces a new direct inversion method for simulating squared Bessel processes by approximating the inverse of the non-central chi-square distribution with a two-dimensional Chebyshev expansion, enabling efficient simulations across various degrees of freedom.

Contribution

The paper presents a novel, accurate, and efficient inversion algorithm for non-central chi-square variables, applicable to squared Bessel and CIR processes, handling small degrees of freedom effectively.

Findings

Accurate simulation of squared Bessel processes across all degrees of freedom.

Efficient generation of noncentral chi-square samples for a range of parameters.

Enhanced simulation of CIR processes using the new inversion method.

Abstract

In this paper we derive a new direct inversion method to simulate squared Bessel processes. Since the transition probability of these processes can be represented by a non-central chi-square distribution, we construct an efficient and accurate algorithm to simulate non-central chi-square variables. In this method, the dimension of the squared Bessel process, equivalently the degrees of freedom of the chi-square distribution, is treated as a variable. We therefore use a two-dimensional Chebyshev expansion to approximate the inverse function of the central chi-square distribution with one variable being the degrees of freedom. The method is accurate and efficient for any value of degrees of freedom including the computationally challenging case of small values. One advantage of the method is that noncentral chi-square samples can be generated for a whole range of values of degrees of freedom using the same Chebyshev coefficients. The squared Bessel process is a building block for the well-known Cox-Ingersoll-Ross (CIR) processes, which can be generated from squared Bessel processes through time change and linear transformation. Our direct inversion method thus allows the efficient and accurate simulation of these processes, which are used as models in a wide variety of applications.