Derivation of the Fractional Fokker–Planck Equation for Stable Lévy with Financial Applications

Abstract

This paper aims to propose a generalized fractional Fokker–Planck equation based on a stable Lévy stochastic process. To develop the general fractional equation, we will use the Lévy process rather than the Brownian motion. Due to the Lévy process, this fractional equation can provide a better description of heavy tails and skewness. The analytical solution is chosen to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. We model market data using a stable distribution to demonstrate the relationships between the tails and the new fractional Fokker–Planck model, as well as develop an R code that can be used to draw figures from real data.