Probability-conservative simulation for Lévy financial model by a mixed finite element method

• Published in: Computers & mathematics with applications (1987) 2022-01, Vol.106, p.92-105
• Main Authors: Chen, Jing, Wang, Feng, Chen, Huanzhen
• Format: Article
• Language: English
• Subjects: Advection; Coercivity; Expanded mixed finite element method; Finite element analysis; Finite element method; Fractional advection diffusion equation; Kurtosis; Lévy financial model; Markov processes; Probability conservative simulation; Solvability and convergence; The inf-sup condition
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2021.12.007

We propose an expanded mixed finite element method (EMFEM) with sharp recognition for such intrinsic natures of a Markov process as the probability conservation, the Chapman-Kolmogorov equation, skewness and excess kurtosis for the fractional-order advection diffusion equation (FADE) oriented in financial engineering. We improve the result in [11] for purely fractional diffusion model by modifying the domain of one bilinear form so that it is well defined for the appearance of the advection term, and the result in [13] for second order elliptic models by using lower-order finite element spaces to reduce computation costs. The solvability and optimal convergence rates of the EMFEM are proved by showing the coerciveness and the inf-sup condition for the modified bilinear forms over the lower-order finite element spaces, and thus the systematically mathematical theory of the EMFEM for general FADEs are developed. Numerical experiments are conducted to confirm these theoretical findings.