An adaptive positive preserving numerical scheme based on splitting method for the solution of the CIR model

Abstract

This paper aims to investigate an adaptive numerical method based on a splitting scheme for the Cox–Ingersoll–Ross (CIR) model. The main challenge associated with numerically simulating the CIR process lies in the fact that most existing numerical methods fail to uphold the positive nature of the solution. Within this article, we present an innovative adaptive splitting scheme. Due to the existence of a square root in the CIR model, the step size is adaptively selected to ensure that, at each step, the value under the square-root does not fall under a given positive level and it is bounded. Moreover, an alternate numerical method is employed if the chosen step size becomes excessively small or the solution derived from the splitting scheme turns negative. This alternative approach, characterized by convergence and positivity preservation, is called the “backstop method”.

Furthermore, we prove the proposed adaptive splitting method ensures the positivity of solutions in the sense that it would be possible to find an interval such that for all stepsizes belong, the probability of using the backstop method can be small. Therefore, the proposed adaptive splitting scheme avoids using the backstop method with arbitrarily high probability. We prove the convergence of the scheme and analyze the convergence rate. Finally, we demonstrate the applicability of the scheme through some numerical simulations, thereby corroborating our theoretical findings.