Positivity-preserving numerical scheme for the alpha-constant elasticity of variance process

Abstract

In this article, we present a method for constructing a positivity-preserving numerical scheme for a jump-extended constant elasticity of variance (CEV) process, where jumps are governed by a spectrally positive α-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 a sufficiently small step size, the proposed scheme converges and theoretically achieves a strong convergence rate of at least $\frac{1}{2}(\frac{{\alpha }_{-}}{2}\wedge \frac{1}{\alpha }\wedge \rho )$, where $\rho \in (\frac{1}{2},1)$ is the Hölder exponent of the jump coefficient $x^{\rho }$ and the constant ${\alpha }_{-}<\alpha$ can be chosen as arbitrarily close to $\alpha \in (1,2)$.