|Unit level:||Level 4|
|Teaching period(s):||Semester 2|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
- MATH20701 - Probability 2 (Compulsory)
Additional RequirementsMATH47112 pre-requisites
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).
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 paying particular attention to the issues of dynamics.
On successful completion of this course unit students will be able to:
- define Gaussian random vectors and processes, compute covariance matrix and drift vector of Gaussian vectors, prove invariance of Gaussian random variables under linear transformations, apply linear transformations to prove that Brownian motion is a Gaussian process;
- define the Brownian motion and outline a proof of existence of the Brownian motion, state and prove the invariance properties of the Brownian motion, state and apply the Law of Large Numbers, the Law of the Itergated Logarithm for the Brownian motion, describe differentiability properties of the Brownian motion;
- define and identify stopping times, define martingales, state and apply Doob’s optional theorem, prove martingale properties of the Brownian motion, apply the martingal properties to prove Wald’s identities;
- define stationary and non-stationary Ornstein-Uhlenbeck process, show that Ornstein- Uhlenbeck process is a Gaussian process and compute its covariance function;
- define the conditional expectation, state basic properties of the conditional expectation, define and apply Markov and strong Markov properties, define Markov, strong Markov and Feller processes, prove that Brownian motion and Ornstein-Uhlenbeck process are strong Markov processes, state and apply the Chapman-Kolmogorov equation;
- define diffusion processes, define and compute infinitesimal generators of Brownian motion, Ornestein-Uhlenbeck process and other diffusion processes, prove backward and forward Kolmogorov equations, derive and apply the Dynkin formula;
- define one-dimensional regular diffusion processes, prove existence of scale function and speed measure, relate scale function and speed measure to infinitesimal generators of diffusion processes;
- relate certain parabolic and elliptic partial differential equations and diffusion processes, find expectation of some functionals of diffusion processes by solving partial differential equations, compute the scale function, speed measure and the Green function of diffusion processes and apply them to find expectation of some functionals of one-dimensional diffusion processes;
- relate optimal stopping problems for diffusion processes and free boundary problems for partial differential equations, solve some classes of partial differential equations to find optimal stopping times and value functions, apply free boundary problems to find the optimal stopping time for American options.
Assessment Further Information
End of semester examination: three hours weighting 100%.
- The heat equation (Fourier's law). [1 lecture]
- The diffusion equation (Fick's law). 
- Einstein's derivation of the diffusion equation (stationary independent increments). 
- The Wiener process (position of a Brownian particle). 
- The Ornstein-Uhlenbeck process (velocity of a Brownian particle). 
- Strong Markov property (starting afresh at stopping times). 
- Diffusion processes (scale function, speed measure, infinitesimal operator). 
- Boundary classification (regular, exit, entrance, natural). 
- The Kolmogorov forward and backward equations. 
- Probabilistic solutions of PDEs (elliptic and parabolic). 
- Optimal stopping, free boundary problems, the American option problem. 
- Optimal stochastic control, the Hamilton-Jacobi-Bellman equation, the optimal consumption-investment problem. 
- Rogers, L. C. G. and Williams, D., Diffusions, Markov Processes and Martingales, Vol. 1 and 2, Cambridge University Press 2000.
- Revuz, D. and Yor, M., Continuous Martingales and Brownian Motion, Springer 1999.
- Karatzas, I. and Shreve, S. E., Brownian Motion and Stochastic Calculus, Springer 1991.
- Karlin, S. and Taylor, H. M., A Second Course in Stochastic Processes, Academic Press 1981.
- Nelson, E., Dynamical Theories of Brownian Motion, Princeton University Press 1967.
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.
- Lectures - 33 hours
- Tutorials - 11 hours
- Independent study hours - 106 hours