Strong Convergence Rates for Cox-Ingersoll-Ross Processes - Full Parameter Range

TL;DR

This paper introduces a Milstein-type approximation scheme for Cox-Ingersoll-Ross processes that achieves polynomial convergence rates across all parameter ranges, including near-zero boundary behavior.

Contribution

It proposes a truncated Milstein scheme for Cox-Ingersoll-Ross processes and proves polynomial convergence rates for the entire parameter spectrum, including boundary regimes.

Findings

Polynomial convergence rates established for all parameter ranges.

Explicit rate for squared Bessel processes of dimension δ>0.

Error measured by maximal Lp-distance on compact intervals.

Abstract

We study strong (pathwise) approximation of Cox-Ingersoll-Ross processes. We propose a Milstein-type scheme that is suitably truncated close to zero, where the diffusion coefficient fails to be locally Lipschitz continuous. For this scheme we prove polynomial convergence rates for the full parameter range including the accessible boundary regime. The error criterion is given by the maximal $L_p$-distance of the solution and its approximation on a compact interval. In the particular case of a squared Bessel process of dimension $\delta>0$ the polynomial convergence rate is given by $\min(1,\delta)/(2p)$.