Stochastic Processes module (MA51005)

Many real-world systems involve randomness: stock markets, weather patterns, population dynamics, and even how particles move through fluids. These are examples of stochastic systems. 

This module gives you the mathematical tools to make sense of those systems using probability, logic, and structured reasoning.

You’ll explore how to model and analyse systems that evolve over time but behave unpredictably. You'll begin with probability spaces and random variables, then move on to Markov chains - systems where the future depends only on the present, not the past. You'll study how these are used to model processes like queues, population changes, or simple games of chance. 

From there, you’ll explore Brownian motion and the mathematics behind random walks, which play a major role in physics and finance. You'll also get an introduction to stochastic differential equations - tools used to model noise and uncertainty in continuous systems. 

This module strengthens your ability to build mathematical models, solve challenging problems, and think rigorously about uncertainty. It’s ideal preparation for careers in data science, research, finance, or applied mathematics. 

What you will learn 

In this module, you will: 

• study random variables, probability measures, and key theorems like the law of large numbers and central limit theorem
• explore Markov chains, including the Markov property, invariant measures, and first passage times
• learn about Brownian motion and how it arises from random walks
• be introduced to stochastic differential equations and Gaussian processes
• apply stochastic models to real-world problems in science, finance, and biology 

By the end of this module, you will be able to: 

• construct and analyse probabilistic models using rigorous mathematical methods
• solve problems involving random walks, Markov chains, and Brownian motion
• apply stochastic techniques to systems with uncertainty
• use analytical and numerical tools to explore randomness in practical settings 

Assignments / assessment

• Coursework (100%)
  • Written problem sets involving theoretical and applied questions 

There is no final exam.  

 Teaching methods / timetable

• Lectures
  • structured exploration of core theory
  • practical examples
• Tutorials
  • small-group problem-solving