First order strong approximation of Ait-Sahalia-type interest rate model with Poisson jumps

TL;DR

This paper introduces a new numerical method, TJABEM, for simulating Ait-Sahalia-type interest rate models with Poisson jumps, achieving strong convergence and domain preservation, supported by theoretical analysis and numerical experiments.

Contribution

The paper develops a transformed jump-adapted backward Euler method that ensures domain preservation and attains strong convergence of order one for complex interest rate models.

Findings

TJABEM preserves the positivity of the interest rate model.

The method achieves a strong convergence rate of order one.

Numerical experiments confirm the theoretical convergence and domain preservation.

Abstract

For Ait-Sahalia-type interest rate model with Poisson jumps, we are interested in strong convergence of a novel time-stepping method, called transformed jump-adapted backward Euler method (TJABEM). Under certain hypothesis, the considered model takes values in positive domain $(0,\infty)$. It is shown that the TJABEM can preserve the domain of the underlying problem. Furthermore, for the above model with non-globally Lipschitz drift and diffusion coefficients, the strong convergence rate of order one of the TJABEM is recovered with respect to a $L^p$-error criterion. Finally, numerical experiments are given to illustrate the theoretical results.