Dynamics of A Degenerate Fokker-Planck Equation and Its Application

Abstract

In this project, a Fokker-Planck equation with two singular points is studied. The equation is derived from a stochastic evolution equation, LMM-SABR model, which is widely used in financial industry. It is difficult to directly study the original equation due to the singularity. As an alternative approach, we introduce appropriate modifications to certain terms of original Fokker-Planck equation at each singular point so that the modified equation has a stationary solution. With the stable stationary solution, the intermediate behavior of the modified Fokker-Planck equation can be captured and described to some extent. The non-modified solutions are compared to modified solutions within finite time and a relatively concrete estimation is given in terms of the modification parameter and the given finite time. We also study some possible modifications. For each modification, the properties of the stationary solution are given. Some numerical results of the time-evolution solutions for these modified equations are also included. As an attempt, we have initiated in this project the study of the difference between the modified and non-modified stochastic differential equations (SDEs). Although no complete analytical results are available, our initial work appears pointing in a promising direction, based on the numerical simulation results that we have observed. The further study of the SDEs will be carried out in the future work.