MSc Pure Mathematics and Mathematical Logic / Course details

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 MATH37001 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.