Explicit modified Euler approximations of the A\"{i}t-Sahalia type model with Poisson jumps

TL;DR

This paper introduces an explicit Euler-type numerical scheme for a complex interest rate model with Poisson jumps, achieving unconditional positivity preservation and a mean-square convergence rate of order 1/2, validated by numerical experiments.

Contribution

A novel explicit Euler scheme with modifications that ensures positivity and achieves the optimal convergence rate for a jump-diffusion interest rate model.

Findings

Scheme preserves positivity unconditionally.

Achieves mean-square convergence rate of 1/2.

Numerical experiments confirm theoretical results.

Abstract

This paper focuses on mean-square approximations of a generalized A\"it-Sahalia interest rate model with Poisson jumps. The main challenge in the construction and analysis of time-discrete numerical schemes is caused by a drift that blows up at the origin, highly nonlinear drift and diffusion coefficients and positivity-preserving requirement. Due to the presence of the Poisson jumps, additional difficulties arise in recovering the exact order $1/2$ of convergence for the time-stepping schemes. By incorporating implicitness in the term $\alpha_{-1}x^{-1} $ and introducing the modifications functions $f_h$ and $g_h$ in the recursion, a novel explicit Euler-type scheme is proposed, which is easy to implement and preserves the positivity of the original model unconditionally, i.e., for any time step-size $h>0$. A mean-square convergence rate of order $1/2$ is established for the proposed scheme in both the non-critical and general critical cases. Finally, numerical experiments are provided to confirm the theoretical findings.