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

TL;DR

This paper introduces an explicit numerical scheme that preserves positivity for complex delay models with jumps, ensuring accurate simulations of financial market processes without explicit solutions.

Contribution

The paper develops a new positivity-preserving numerical method for CIR/CEV delay models with jumps, with proven convergence and minimal numerical validation.

Findings

The scheme maintains non-negativity of solutions.

The method converges strongly to the true solution.

Numerical experiments support the scheme's effectiveness.

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.