Stochastic Processes

This repository contains MATLAB, Python, and Jupyter notebook resources for exploring and simulating stochastic processes.
It includes theoretical studies, numerical simulations, and comparisons between stochastic integration methods for dynamical systems.

📁 Repository Structure and Contents

| File | Description | |:-----|:------------| | CubicgeodesicMPC.ipynb | Exploration of cubic geodesic paths versus Laplace assumptions in Model Predictive Control (MPC). | | CubicvsLA.ipynb | Comparison between cubic paths and Laplace approximations. | | KramersSS.m | MATLAB script for steady-state analysis of Kramers equation. | | Kramers_euler_maruyama.m | MATLAB simulation of Kramers dynamics with Euler-Maruyama method. | | Kramersequation.ipynb | Jupyter notebook exploring stochastic Kramers equation. | | Morris_Lecar_SS.ipynb | Steady-state and stochastic analysis of Morris-Lecar neuron model. | | OUEulerMaruyama.py | Python simulation of an Ornstein-Uhlenbeck process using Euler-Maruyama method. | | OUProcessSS.m | MATLAB steady-state analysis of an Ornstein-Uhlenbeck process. | | OUdeterministicsimu.ipynb | Deterministic simulation of Ornstein-Uhlenbeck dynamics. | | OUprocess.py | Python module for Ornstein-Uhlenbeck process simulations. | | Pendulumwithfriction.ipynb | Stochastic simulation of a pendulum system with friction. | | cubiclaplaceassum.m | MATLAB function related to cubic approximation or Laplace assumptions. | | lorentzstochastic.ipynb | Stochastic simulation of the Lorenz system. | | ornstein_uhlenbeck_euler_maruyama.m | MATLAB implementation of Euler-Maruyama simulation for OU process. | | ouMilstein_py.ipynb | Python notebook applying Milstein's method for stochastic differential equations. | | stochasticsimusOU.ipynb | Collection of Ornstein-Uhlenbeck stochastic simulations. | | citation.cff | Citation file for properly referencing this work. |

📚 Topics Covered

• Ornstein-Uhlenbeck processes (stochastic and deterministic)
• Kramers equation and simulations
• Euler-Maruyama and Milstein methods for SDEs
• Stochastic modeling of mechanical systems (e.g., pendulum with friction, Lorenz attractor)
• Neuron model dynamics with stochasticity (Morris-Lecar model)
• Model Predictive Control concepts linked with stochastic approximations

🛠 Requirements

• MATLAB R2020b or newer
• Python 3.8+
• Key Python libraries:
  • numpy
  • matplotlib
  • scipy
  • sympy (for some symbolic computations)
  • jupyter for notebooks

Install Python libraries with: bash pip install numpy matplotlib scipy sympy jupyter

🚀 How to Use

1. Clone the repository: bash git clone https://github.com/YOUR_USERNAME/StochasticProcesses.git
2. Open Jupyter Notebooks for Python-based simulations: bash jupyter notebook
3. Run MATLAB scripts directly for theoretical and numerical analysis.

✨ Highlights

• Cross-discipline study of stochastic processes (physics, biology, control).
• Hands-on simulation of stochastic differential equations.
• Blending deterministic and stochastic dynamics in modeling.

📜 License

This project is licensed under the MIT License — see the LICENSE file if available.

👨‍💻 Author

Developed by Adrian Guel.