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.

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

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.

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.