A High Order Finite Difference Method for Tempered Fractional Diffusion Equations with Applications to the CGMY Model

Abstract

Using the weighted and shifted Lubich difference (WSLD) operator, we propose an efficient and stable difference scheme for solving space tempered fractional diffusion equations. The scheme has fourth order accuracy in space and second order accuracy in time. To improve the computational efficiency further, a fourth order backward Euler scheme is used to achieve fourth order accuracy in time. The fourth order backward Euler method is shown to be stable under a minor requirement. For two-dimensional problems, the Crank--Nicolson alternating direction implicit (CN-ADI) scheme is used to reduce computational complexity. These proposed methods are applied to the Carr--Geman--Madan--Yor (CGMY) model in finance, and several numerical experiments are designed to verify the efficiency of the proposed methods.