Stochastic Density Dynamics

Visualising the Fokker-Planck equation through Monte Carlo simulation of the Ornstein-Uhlenbeck process.

Overview

This project explores the relationship between discrete stochastic processes and their continuous probability density evolution. By simulating thousands of particles following the Ornstein-Uhlenbeck (OU) mean-reverting process, we reconstruct the solution to the Fokker-Planck equation and visualise it as a 3D probability surface.

The Maths

Stochastic Differential Equation (SDE):

$$dX_t = \theta(\mu - X_t)dt + \sigma dW_t$$

Fokker-Planck Equation (PDE):

$$\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}\left[ \theta(\mu - x)p \right] + \frac{\partial^2}{\partial x^2}\left[ \frac{\sigma^2}{2} p \right]$$

Dependencies

• NumPy
• Matplotlib
• SciPy

Author

April Kidd