A positivity preserving numerical scheme for the mean-reverting alpha-CEV process

In this article, we present a method to construct a positivity preserving numerical scheme for both the jump-extended CEV process and jump-extended CIR process, whose jumps are governed by a (compensated) spectrally positive $\alpha$-stable process with $\alpha \in (1,2)$. The proposed scheme is obtained by making the diffusion coefficient partially implicit and then finding the appropriate adjustment factor. We show that the proposed scheme converges and theoretically achieves a strong convergence rate of at least $\frac{1}{2}\left(\frac{\alpha_-}{2} \wedge \frac{1}{\alpha}\right)$, where the constant $\alpha_- < \alpha$ can be chosen arbitrarily close to $\alpha \in (1,2)$. Finally, to support our result, we present some numerical simulations which suggest that the optimal rate of convergence is $\frac{\alpha_-}{4}$.