Convergence of a splitting method for a general interest rate model

TL;DR

This paper introduces and proves the mean-square convergence of a novel tamed-splitting numerical method for a generalized Ait-Sahalia interest rate model, effectively handling non-globally Lipschitz coefficients.

Contribution

The paper develops a new tamed-splitting method with a backstop to ensure positivity, proving its convergence and small probability of requiring the backstop, advancing numerical solutions for interest rate models.

Findings

Achieves mean-square convergence rate of order one.

Ensures the probability of the backstop being used is arbitrarily small.

Demonstrates effectiveness through numerical experiments.

Abstract

We prove mean-square convergence of a novel numerical method, the tamed-splitting method, for a generalized Ait-Sahalia interest rate model. The method is based on a Lamperti transform, splitting and applying a tamed numerical method for the nonlinearity. The main difficulty in the analysis is caused by the non-globally Lipschitz drift coefficients of the model. We examine the existence, uniqueness of the solution and boundedness of moments for the transformed SDE.We then prove bounded moments and inverses moments for the numerical approximation. The tamed-splitting method is a hybrid method in the sense that a backstop method is invoked to prevent solutions from overshooting zero and becoming negative. We successfully recover the mean-square convergence rate of order one for the tamed-splitting method. In addition we prove that the probability of ever needing the backstop method to prevent a negative value can be made arbitrarily small. In our numerical experiments we compare to other numerical methods in the literature for realistic parameter values.