On the backward Euler method for a generalized Ait-Sahalia-type rate model with Poisson jumps
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On the backward Euler method for a generalized Ait-Sahalia-type rate model with Poisson jumps Yuying Zhao1 · Xiaojie Wang1 · Mengchao Wang1 Received: 14 January 2020 / Accepted: 18 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This article aims to reveal the mean-square convergence rate of the backward Euler method (BEM) for a generalized Ait-Sahalia interest rate model with Poisson jumps. The main difficulty in the analysis is caused by the non-globally Lipschitz drift and diffusion coefficients of the model. We show that the BEM preserves the positivity of the original problem. Furthermore, we successfully recover the mean-square convergence rate of order one-half for the BEM. The theoretical findings are accompanied by several numerical examples. Keywords Ait-Sahalia model · Poisson jumps · Backward Euler method · Mean-square convergence rate Mathematics Subject Classification (2010) 60H35 · 60H15 · 65C30
1 Introduction As is well known, stochastic differential equations (SDEs) are widely used in various scientific areas to model real-life phenomena affected by random noises. However, in
This work was supported by Natural Science Foundation of China (12071488, 11671405, 11971488, 91630312), Innovation Program of Central South University(No.2019zzts397), and Natural Science Foundation of Hunan Province for Distinguished Young Scholars (2020JJ2040). Mengchao Wang
[email protected] Yuying Zhao [email protected] Xiaojie Wang [email protected] 1
School of Mathematics and Statistics, Central South University, Changsha, China
Numerical Algorithms
order to model the event-driven phenomena, it is necessary and significant to introduce SDEs with Poisson jumps [6, 24]. For instance, the stock price movements might suffer from sudden and significant impacts caused by unpredictable important events such as market crashes, announcements made by central banks, and changes in credit rating. Over the last decade, SDEs with jumps have become increasingly popular for modelling market fluctuations, both for risk management and option pricing purposes (see, e.g., [6]). Since the analytic solutions of nonlinear SDEs with jumps are rarely available, numerical approximations become a powerful tool to understand the behavior of the underlying problems. Motivated by this, a great deal of research papers are devoted to the interesting topic (see, e.g., [2–5, 7, 9, 11–13, 16–19, 24, 25, 27, 29]). The present article is concerned with the mean-square convergence analysis of a time-stepping scheme for a generalized Ait-Sahalia interest rate model with Poisson jumps, which takes the form as follows: dXt = (a−1 Xt−1 −a0 +a1 Xt −a2 Xt ) dt +bXtθ dWt +ϕ(Xt − ) dNt , γ
X0 = x0 ,
t > 0. (1) Here, a−1 , a0 , a1 , a2 , b > 0, θ , γ > 1, ϕ : R → R, and Xt − := lims→t − Xs . Moreover, we let {Wt }t∈[0,∞) and {Nt }t∈[0,∞) be a one-dimensional Brownian motion and a Poisson process with the jump intensity λ > 0, respectively, on a filtered probability space (, F , P, {Ft }t≥0 ) with r
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