  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.
