Communications in Mathematical Sciences

Volume 22 (2024)

Number 6

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

Pages: 1569 – 1599

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n6.a6

Authors

Wansheng Wang (Department of Mathematics, Shanghai Normal University, Shanghai, China)

Mengli Mao (Department of Mathematics, Shanghai Normal University, Shanghai, China)

Zifeng Li (Department of Mathematics, Shanghai Normal University, Shanghai, China)

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.

Keywords

parabolic integro-differential equations, IMEX Runge-Kutta methods, stage-based interpolation Runge-Kutta methods, option pricing models, nonsmooth initial data, stability, error estimates

2010 Mathematics Subject Classification

Primary 65L06, 65M06, 65M12. Secondary 65J10, 91B24, 91G60.

Received 19 January 2023

Received revised 13 September 2023

Accepted 7 January 2024

Published 18 July 2024