# B8.1 Probability, Measure and Martingales (2018-2019)

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Part A Integration is a prerequisite, so that the corresponding material will be assumed to be known. Part A Probability is a prerequisite.

16 lectures

### Assessment type:

- Written Examination

Probability theory arises in the modelling of a variety of systems where the understanding of the "unknown'' plays a key role, such as population genetics in biology, market evolution in financial mathematics, and learning features in game theory. It is also very useful in various areas of mathematics, including number theory and partial differential equations. The course introduces the basic mathematical framework underlying its rigorous analysis, and is therefore meant to provide some of the tools which will be used in more advanced courses in mathematics.

The first part of the course provides a review of measure theory from Integration Part A, and develops a deeper framework for its study. Then we proceed to develop notions of conditional expectation, martingales, and to show limit results for the behaviour of these martingales which apply in a variety of contexts.

The students will learn about measure theory, random variables, independence, expectation and conditional expectation, product measures and discrete-parameter martingales.

Review of $ \sigma $-algebras, measure spaces. Uniqueness of extension of $ \pi $-systems and Carathéodory's Extension Theorem, monotone-convergence properties of measures, $ \limsup $ and $ \liminf $ of a sequence of events, Fatou's Lemma, reverse Fatou Lemma, first Borel-Cantelli Lemma.

Random variables and their distribution functions, $ \sigma $-algebras generated by a collection of random variables. Product spaces. Independence of events, random variables and $ \sigma $-algebras, $ \pi $-systems criterion for independence, second Borel-Cantelli Lemma. The tail $ \sigma $-algebra, Kolomogorov's 0-1 Law. Convergence in measure and convergence almost everywhere.

Integration and expectation, review of elementary properties of the integral and $ L^p $ spaces [from Part A Integration for the Lebesgue measure on $ \mathbb{R} $]. Scheffé's Lemma, Jensen's inequality. The Radon-Nikodym Theorem [without proof]. Existence and uniqueness of conditional expectation, elementary properties. Relationship to orthogonal projection in $ L^2 $.

Filtrations, martingales, stopping times, discrete stochastic integrals, Doob's Optional-Stopping Theorem, Doob's Upcrossing Lemma and "Forward'' Convergence Theorem, martingales bounded in $ L^2 $, Doob decomposition, Doob's submartingale inequalities.

Uniform integrability and $ L^1 $ convergence, backwards martingales and Kolmogorov's Strong Law of Large Numbers.

Examples and applications.

- D. Williams,
*Probability with Martingales*, Cambridge University Press, 1995. - Lecture Notes for the course.

*Please note that e-book versions of many books in the reading lists can be found on SOLO and ORLO.*

- Z. Brzezniak and T. Zastawniak, Basic stochastic processes. A course through exercises. Springer Undergraduate Mathematics Series. (Springer-Verlag London, Ltd., 1999) [more elementary than D. Williams' book, but can provide with a complementary first reading].
- M. Capinski and E. Kopp,
*Measure, integral and probability*, Springer Undergraduate Mathematics Series. (Springer-Verlag London, Ltd., second edition, 2004). - R. Durrett,
*Probability: Theory and Examples*(Second Edition Duxbury Press, Wadsworth Publishing Company, 1996). - A. Etheridge,
*A Course in Financial Calculus*, (Cambridge University Press, 2002). - J. Neveu,
*Discrete-parameter Martingales*(North-Holland, Amsterdam, 1975).