ST453      Half Unit
Probability and Mathematical Statistics II

This information is for the 2021/22 session.

Teacher responsible

Prof Konstantinos Kardaras


This course is available on the MSc in Quantitative Methods for Risk Management. This course is available with permission as an outside option to students on other programmes where regulations permit.


Probability and Mathematical Statistics I is a pre-requisite.

Course content

This course provides instruction in advanced topics in probability and mathematical statistics, mainly based on

martingale theory. It is a continuation of  Probability and Mathematical Statistics I. The following topics will in particular be covered:

  1. Conditional expectation revisited; linear regression; martingales and first examples.
  2. Concentration inequalities; dimension reduction; log-Sobolev inequalities.
  3. Martingale transforms; optional sampling theorem; convergence theorems.
  4. Sequential testing; backwards martingales; law of large numbers; de Finetti’s theorem.
  5. Markov chains; recurrence; reversibility; foundations of MCMC.
  6. Ergodic theory.
  7. Brownian motion; quadratic variation; stochastic integration.
  8. Stochastic differential equations; diffusions; filtering.
  9. Bayesian updating; Ergodic diffusions; Langevin samplers.
  10. Brownian bridge; empirical processes; Kolmogorov-Smirnov statistic.


This course will be delivered through a combination of classes, lectures and Q&A sessions totalling a minimum of 30 hours across Lent Term. This year, some of this teaching may be delivered through a combination of classes and flipped-lectures delivered as short online videos. This course includes a reading week in Week 6 of Lent Term.

Formative coursework

Students will be expected to produce 9 problem sets in the LT.

Weekly problem sets that are discussed in subsequent seminars. The coursework that will be used for summative assessment will be chosen from a subset of these problems.

Indicative reading

  1. Williams, D. (1991). Probability with Martingales. Cambridge University Press.
  2. Durrett, R. (2019). Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics.
  3. Karatzas, I, Shreve S. (1991). Brownian motion and Stochastic Calculus. Springer GTM.
  4. Shao, J. (2007). Mathematical Statistics. Springer Texts in Statistics.
  5. Keener, R. (2010). Theoretical Statistics. Springer Texts in Statistics.


Exam (70%, duration: 2 hours, reading time: 10 minutes) in the summer exam period.
Coursework (30%) in the LT.

Three of the homework problem sets will be submitted and marked as assessed coursework.

Course selection videos

Some departments have produced short videos to introduce their courses. Please refer to the course selection videos index page for further information.

Important information in response to COVID-19

Please note that during 2021/22 academic year some variation to teaching and learning activities may be required to respond to changes in public health advice and/or to account for the differing needs of students in attendance on campus and those who might be studying online. For example, this may involve changes to the mode of teaching delivery and/or the format or weighting of assessments. Changes will only be made if required and students will be notified about any changes to teaching or assessment plans at the earliest opportunity.

Key facts

Department: Statistics

Total students 2020/21: 2

Average class size 2020/21: 2

Controlled access 2020/21: Yes

Value: Half Unit

Guidelines for interpreting course guide information

Personal development skills

  • Problem solving
  • Application of numeracy skills
  • Specialist skills