# Non-Analytic Solution to the Fokker-Planck Equation of Fractional Brownian Motion via Laplace Transforms

## TL;DR

This paper derives a non-analytic solution to the Fokker-Planck equation for fractional Brownian motion using Laplace transforms, enabling the calculation of transition probabilities and application to financial models.

## Contribution

It introduces a novel non-analytic solution method for the Fokker-Planck equation of fractional Brownian motion and applies it to the Cox-Ingersoll-Ross model.

## Key findings

- Derived the transition probability density function for fractional Brownian motion.
- Applied the solution to the Cox-Ingersoll-Ross model with fractional Brownian motion.
- Provided a new approach for analyzing stochastic differential equations with fractional noise.

## Abstract

This paper derives the non-analytic solution to the Fokker-Planck equation of fractional Brownian motion using the method of Laplace transform. Sequentially, by considering the fundamental solution of the non-analytic solution, this paper obtains the transition probability density function of the random variable that is described by the It\^o's stochastic ordinary differential equation of fractional Brownian motion. Furthermore, this paper applies the derived transition probability density function to the Cox-Ingersoll-Ross model governed by the fractional Brownian motion instead of the usual Brownian motion.