IMEX variable step-size Runge-Kutta methods for parabolic integro-differential equations with nonsmooth initial data

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-18 DOI:10.4310/cms.2024.v22.n6.a6

Wansheng Wang, Mengli Mao, Zifeng Li

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引用次数: 0

Abstract

We develop a class of implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving parabolic integro-differential equations (PIDEs) with nonsmooth initial data, which describe several option pricing models in mathematical finance. Different from the usual IMEX RK methods, the proposed methods approximate the integral term explicitly by using an extrapolation operator based on the stage-values of RK methods, and we call them as IMEX stage-based interpolation RK (SBIRK) methods. It is shown that there exist arbitrarily high order IMEX SBIRK methods which are stable for abstract PIDEs under suitable time step restrictions. The consistency error and the global error bounds for this class of IMEX Runge-Kutta methods are derived for abstract PIDEs with nonsmooth initial data. The related higher time regularity analysis of the exact solution and stability estimates for IMEX SBIRK methods play key roles in deriving these error bounds. Numerical experiments for European options under jump-diffusion models and stochastic volatility model with jump verify and complement our theoretical results.

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具有非光滑初始数据的抛物整微分方程的 IMEX 可变步长 Runge-Kutta 方法

我们开发了一类隐式显式（IMEX）Runge-Kutta（RK）方法，用于求解具有非光滑初始数据的抛物线整微分方程（PIDE），该方程描述了数学金融中的若干期权定价模型。与通常的 IMEX RK 方法不同，所提出的方法通过使用基于 RK 方法阶段值的外推算子来显式逼近积分项，我们称之为基于阶段值的 IMEX 插值 RK（SBIRK）方法。研究表明，存在任意高阶的 IMEX SBIRK 方法，这些方法在合适的时间步长限制下对抽象 PIDE 是稳定的。对于具有非光滑初始数据的抽象 PIDE，推导出了该类 IMEX Runge-Kutta 方法的一致性误差和全局误差边界。精确解的相关高时间正则性分析和 IMEX SBIRK 方法的稳定性估计在推导这些误差边界时发挥了关键作用。在跳跃-扩散模型和带有跳跃的随机波动模型下对欧式期权进行的数值实验验证并补充了我们的理论结果。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.