MSc Pure Mathematics

For entry in the academic year beginning September 2026, the tuition fees are as follows:

• MSc (full-time)
  UK students (per annum): £14,700
  International, including EU, students (per annum): £36,300
• MSc (part-time)
  UK students (per annum): £7,400
  International, including EU, students (per annum): £14,700

Further information for EU students can be found on our dedicated EU page.

All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).

The Martingale Foundation helps provide access to postgraduate mathematics study for UK students facing financial barriers by offering fully-funded MSc and PhD programmes. Find more information on the Martingale Foundation website . The deadline is Sunday 19 October 2025.

Our Manchester Master's Bursaries are aimed at widening access to master's courses by removing barriers to postgraduate education for students from underrepresented groups.

We also welcome the best and brightest international students each year and reward excellence with a number of merit-based scholarships. See our range of master’s scholarships for international students .

If you have completed or are in the final year of an undergraduate degree at The University of Manchester, you may be eligible for a discount of 10% on tuition fees for postgraduate study. Find out if you're eligible and how to apply .

For more information on master's tuition fees and studying costs, visit the University of Manchester funding for master's courses website.

Brownian motion is the most important stochastic process. It was observed by Brown in 1828 and explained by Einstein in 1905. A more accurate model based on work of Langevin from 1908 was introduced by Ornstein and Uhlenbeck in 1930. The assumption of stationary independent increments made by Einstein in 1905 has had a profound influence on the development of probability theory in the 20th century. The course unit presents basic facts and ideas of Brownian motion and one-dimensional diffusion processes
 

Pre-requisites: MATH47201/67201 or MATH37002 or MATH47101/67101

Students are not permitted to take, for credit, MATH47112 in an undergraduate programme and then MATH67112 in a postgraduate programme at the University of Manchester, as the courses are identical.

The unit aims to provide the basic knowledge necessary to pursue further studies/applications where Brownian motion plays a fundamental role (e.g. Financial Mathematics).

On successful completion of this course unit students will be able to: 

• define Gaussian random vectors and processes and prove their basic properties and calculate their basic characteristics 
   
• define Brownian motion, Ornstein-Uhlenbeck process and related processes, and derive and apply some basic properties and limit theorems.
   
• define and apply fundamental theorems for stopping times and martingales. 
   
• define Markov and Feller processes and one-dimensional regular diffusions, and prove and apply some basic properties of these processes.
   
• define and apply scale function, speed measure, infinitesimal generators and prove their basic properties, and prove and apply backward and forward Kolmogorov equations and Dynkin’s formula.
   
• relate (free) boundary problems for certain parabolic and elliptic partial differential equations to (optimal) stopping problems for diffusions. 
   

 

Syllabus:

• Gaussian vectors [1] 
   
• Brownian motion (definition, existence and basic properties) [2]
   
• Martingale and Markov properties of Brownian motion [4]
   
• Markov processes, strong Markov processes and Feller processes [4]
   
• One dimensional diffusion processes (scale function, Green function, speed measure, infinitesimal generator). [6]
   
• Probabilistic solutions of PDEs (elliptic and parabolic). [3]
   
• Optimal stopping, free boundary problems, the American option problem. [2]
   

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

1. Schilling, R.L and Partzsch, L. Brownian Motion: An Introduction to Stochastic Processes, De Gruyter 2012. 
2. Breiman, L. Probability. Siam, 1992. 
3. Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, Springer 1999.
4. Rogers, L. C. G. and Williams, D., Diffusions, Markov Processes and Martingales, Vol. 1 and 2, Cambridge University Press 2000.
5. Karlin, S. and Taylor, H. M., A Second Course in Stochastic Processes, Academic Press 1981.
6. Mörters, P and Peres, Y. Cambridge University Press 2010.