An explicit positivity preserving numerical scheme for CIR/CEV type delay models with jump

Abstract

We consider mean-reverting CIR/CEV processes with delay and jumps used as models on the financial markets. These processes are solutions of stochastic differential equations with jumps, which have no explicit solutions. We prove the non-negativity property of the solution of the above models and propose an explicit positivity preserving numerical scheme,using the semi-discrete method, that converges in the strong sense to the exact solution. We also make some minimal numerical experiments to illustrate the proposed method.