# B8.1 Probability, Measure and Martingales - Material for the year 2020-2021

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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 strongly recommended.

16 lectures

### Assessment type:

- Written Examination

Probability is both a fundamental way of viewing the world and a core mathematical discipline. In recent years there has been an explosive growth in the importance of probability in scientific research. Applications range from physics to neuroscience, from genetics to communication networks and, of course, finance.

This course develops the mathematical foundations essential for more advanced courses in probability theory, and introduces the notion of martingales. The first part of the course develops a more sophisticated understanding of the measure theory in Part A Integration. We then introduce conditional expectation and martingales, and establish some results that allow us to deduce their longterm behaviour.

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.

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

- 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). - J. Neveu,
*Discrete-parameter Martingales*(North-Holland, Amsterdam, 1975).