Fokker-Planck equation driven by asymmetric Lévy motion

Abstract

Non-Gaussian Lévy noises are present in many models for understanding underlining principles of physics, finance, biology and more. In this work, we consider the Fokker-Planck equation(FPE) due to one-dimensional asymmetric Lévy motion, which is a nonlocal partial differential equation. We present an accurate numerical quadrature for the singular integrals in the nonlocal FPE and develop a fast summation method to reduce the order of the complexity from O(J 2 ) to O(J log J) in one time-step, where J is the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises and the domain size.

HJB and Fokker-Planck equations for river environmental management based on stochastic impulse control with discrete and random observation

“…A more sophisticated numerical scheme can be more complicated, but may be able to achieve higher-order convergence. The FP equation was found to be discontinuous even in the simplified case, implying that schemes naturally allowing for solution discontinuities, such as the discontinuous Galerkin methods [83][84], can be a more accurate alternative to our cell-centered FV scheme. For both HJB and FP equations, utilizing some moving mesh methods, like the Moving Mesh Partial Differential Equation methods [85][86], can also be an alternative to accurately resolve discontinuous solutions.…”

Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative $α$-stable noises

“…For α = 0.5, it is skew to right when β = −0.5, but it is opposite when β = 0.5, which is also different for α = 1.5. It is interesting to see that the the solution of FPE is asymmetric for multiplicative Lévy noise, while it is symmetric for additive case (see [28]).…”