Nonlinear Fokker-Planck Equation in the Model of Asset Returns

TL;DR

This paper analyzes a nonlinear Fokker-Planck equation with specific coefficients and nonlocal terms within a financial asset returns model, providing exact solutions for special cases and asymptotic solutions in general.

Contribution

It introduces a novel approach to solving a nonlinear Fokker-Planck equation in financial modeling, including exact solutions and semiclassical asymptotic approximations.

Findings

Exact solutions for special cases of the equation.

Asymptotic solutions using the WKB-Maslov method.

Application to modeling asset return distributions.

Abstract

The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leading term of the Cauchy problem solution asymptotic in a formal small parameter in semiclassical approximation following the complex WKB-Maslov method in the class of trajectory concentrated functions.