2012 Young Researchers Workshop:
Kinetic Description of Multiscale Phenomena

Application of Deterministic Particle Methods to PDEs Arising in Financial Modeling

Shumo Cui

Tulane University


We numerically study convection-diffusion equations arising in financial modeling. We focus on the convection-dominated case, in which the diffusion coefficients are small. Both inite-difference and Monte-Carlo methods which are wildly used in the problems of this kind are typically inefficient due to severe restrictions on the meshsize and the number of realizations needed to achieve high resolution. We develop new deterministic particle methods which introduce very small numerical diffusion and do not suffer from the aforementioned drawbacks. Our approach is based on the operator splitting: The hyperbolic steps are made using the method of characteristics, while the parabolic steps are performed using a special discretization of the integral representation of the solution. We apply our scheme to a variety of test problems and the numerical results clearly demonstrate a high accuracy, efficiency and robustness of the proposed method. This is a joint work with Alexander Kurganov (Tulane University) and Alexei Medovikov (Susquehanna International Group).