On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case

TL;DR

This paper introduces a positivity-preserving implicit Euler-Maruyama scheme for a jump-extended CIR process driven by an alpha-stable process with infinite activity jumps, providing convergence analysis and numerical illustrations.

Contribution

It develops a novel positivity-preserving numerical scheme for jump-extended CIR processes with infinite activity jumps driven by alpha-stable processes.

Findings

Established the strong convergence rate of the scheme.

Provided numerical examples demonstrating the scheme's effectiveness.

Extended the applicability of positivity-preserving schemes to infinite activity jump processes.

Abstract

We propose a positivity preserving implicit Euler-Maruyama scheme for a jump-extended Cox-Ingersoll-Ross (CIR) process where the jumps are governed by a compensated spectrally positive $\alpha$-stable process for $\alpha \in (1,2)$. Different to the existing positivity preserving numerical schemes for jump-extended CIR or CEV (Constant Elasticity Variance) process, the model considered here has infinite activity jumps. We calculate, in this specific model, the strong rate of convergence and give some numerical illustrations. Jump extended models of this type were initially studied in the context of branching processes and was recently introduced to the financial mathematics literature to model sovereign interest rates, power and energy markets.