A positivity preserving numerical scheme for the alpha-CEV process

TL;DR

This paper introduces a positivity-preserving numerical scheme for a jump-extended CEV process driven by spectrally positive alpha-stable jumps, ensuring convergence and stability for small step sizes.

Contribution

It develops a novel partially implicit scheme for the alpha-CEV process that guarantees positivity and achieves a proven strong convergence rate.

Findings

The scheme preserves positivity under small step sizes.

Convergence rate is at least half, depending on alpha and the H"older exponent.

The method is applicable to jump processes with spectrally positive alpha-stable jumps.

Abstract

In this article, we present a method to construct a positivity-preserving numerical scheme for a jump-extended CEV (Constant Elasticity of Variance) process, whose jumps are governed by a spectrally positive $\alpha$-stable process with $\alpha \in (1,2)$. The numerical scheme is obtained by making the diffusion coefficient $x^\gamma$, where $\gamma \in (\frac{1}{2},1)$, partially implicit and then finding the appropriate adjustment factor. We show that, for sufficiently small step size, the proposed scheme converges and theoretically achieves a strong convergence rate of at least $\frac{1}{2}\left(\frac{\alpha_-}{2} \wedge \frac{1}{\alpha}\wedge \rho\right)$, where $\rho \in (\frac{1}{2},1)$ is the H\"older exponent of the jump coefficient $x^\rho$ and the constant $\alpha_- < \alpha$ can be chosen arbitrarily close to $\alpha \in (1,2)$.