Positivity-Preserving Explicit Approximations of the Aït-Sahalia Interest Rate Model with Poisson Jumps

Abstract

This paper focuses on positivity-preserving approximations of a generalized Aït-Sahalia interest rate model with Poisson jumps. In addition to the challenge caused by a drift that blows up at the origin, the highly nonlinear diffusion coefficients and the positivity-preserving requirement, the current study faces essential difficulties due to the presence of the Poisson jumps. By incorporating implicitness in the term

{\alpha }_{-1}{x}^{-1}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha _{-1}x^{-1} $$\end{document}

and introducing some modification functions

f_h

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_h$$\end{document}

and

g_h

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$g_h$$\end{document}

, a novel family of explicit Euler-type schemes 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

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$h>0$$\end{document}

. 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.

Keywords

Aït-Sahalia model with Poisson jumps / Unconditionally positivity-preserving / Explicit Euler-type scheme / Mean-square convergence rate / 60H35 / 60H15 / 65C30

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Funding

National Natural Science Foundation of China(12471394)

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Shanghai University

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