An adaptive splitting method for the Cox-Ingersoll-Ross process

TL;DR

This paper introduces a new adaptive splitting method for numerically solving the Cox-Ingersoll-Ross process, achieving improved convergence rates and stability across various parameter regimes.

Contribution

The paper presents a novel splitting scheme with proven strong error bounds and extends it to all parameter regimes using a hybrid approach with a soft zero region.

Findings

Strong error order of 1/4 in certain regimes

Numerical rate of order 1 observed in practice

Adaptive method shows smaller error constants at large noise

Abstract

We propose a new splitting method for strong numerical solution of the Cox-Ingersoll-Ross model. For this method, applied over both deterministic and adaptive random meshes, we prove a uniform moment bound and strong error results of order $1/4$ in $L_1$ and $L_2$ for the parameter regime $\kappa\theta>\sigma^2$. We then extend the new method to cover all parameter values by introducing a \emph{soft zero} region (where the deterministic flow determines the approximation) giving a hybrid type method to deal with the reflecting boundary. From numerical simulations we observe a rate of order $1$ when $\kappa\theta>\sigma^2$ rather than $1/4$. Asymptotically, for large noise, we observe that the rates of convergence decrease similarly to those of other schemes but that the proposed method making use of adaptive timestepping displays smaller error constants.