A positivity preserving numerical scheme for the alpha-CEV process

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 $\hat{I}\pm$-stable process with $\hat{I}\pm \in (1,2)$. The numerical scheme is obtained by making the diffusion coefficient $x^3$, where ${\hat{I}}^3\in (\frac{1}{2},1)$, 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 $\frac{1}{2}\left(\frac{\hat{I}{\pm }_{-}}{2}\wedge \frac{1}{\hat{I}}\pm \wedge \ddot{I}\right)$, where $\ddot{I}\in (\frac{1}{2},1)$ is the Hölder exponent of the jump coefficient $x^{\ddot{I}}$ and the constant $\hat{I}{\pm }_{-}<\hat{I}\pm$ can be chosen arbitrarily close to $\hat{I}\pm \in (1,2)$.