- Lecturer: Jan Obloj
General Prerequisites:
The course is self-contained but subsumes both Part A Probability and Part A Integration and relies strongly on the intuition and knowledge build up in those two courses. Consequently, both are very strongly recommended.
Course Term: Michaelmas
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: H
Assessment Type: Written Examination
Course Overview:
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. The first part of the course develops a more sophisticated understanding of measure theory and integration, first seen in Part A Integration. The second part focuses on key probabilistic concepts: independence and conditional expectation. We then introduce discrete time martingales and establish results needed to study their behaviour. This prepares the ground for continuous martingales, studied in B8.2, which are the cornerstone of stochastic calculus.
This course develops the mathematical foundations essential for more advanced courses in probability theory. The first part of the course develops a more sophisticated understanding of measure theory and integration, first seen in Part A Integration. The second part focuses on key probabilistic concepts: independence and conditional expectation. We then introduce discrete time martingales and establish results needed to study their behaviour. This prepares the ground for continuous martingales, studied in B8.2, which are the cornerstone of stochastic calculus.
Learning Outcomes:
The students will learn about measure theory, random variables, independence, expectation and conditional expectation, product measures, filtrations and stopping times, discrete-parameter martingales and their applications.
Course Synopsis:
Measurable sets, \(\sigma\)-algebras, \(\pi\)-\(\lambda\) systems lemma. Random variables, generated σ-algebras, monotone class theorem. Measures: properties, uniqueness of extension, Carathéodory's Extension Theorem; measure spaces, pushforward measure, product measure.
Independence of events, random variables and \(\sigma\)-algebras, relation to product measures.
The tail \(\sigma\)-algebra, Kolomogorov's 0-1 Law, lim sup and lim inf of a sequence of events, Fatou and reverse Fatou Lemma for sets, Borel-Cantelli Lemmas.
Integration and expectation, review and extension of elementary properties of the integral and convergence theorems [from Part A Integration for the Lebesgue measure on \(R\)]. Radon-Nikodym Theorem [without proof], Scheffé's Lemma. Integration on product space, Fubini/Tonelli Theorem. Different modes of convergence and their relations. Markov’s and Jensen’s inequalities. \(L^p\) spaces, Holder’s and Minkowski’s inequalities, completeness. Uniform integrability, Vitali’s convergence theorem.
Conditional expectation: definition, properties, uniquness. Conditional convergence theorems and inequalities, link with uniform integrability. Orthogonal projection in \(L^2\), existence of conditional expectation.
Filtrations and stopping times. Examples and properties. \(\sigma\)-algebra associated to a stopping time.
Martingales in discrete time: definition, examples, properties, discrete stochastic integrals. Doob’s decomposition theorem. Stopped martingales and Doob's Optional Sampling Theorem. Maximal and \(L^p\) Inequalities, Doob's Upcrossing Lemma and Martingale Convergence Theorem. Uniformly integrable martingales, convergence in \(L^1\). Backwards martingales and Kolmogorov's Strong Law of Large Numbers.
Independence of events, random variables and \(\sigma\)-algebras, relation to product measures.
The tail \(\sigma\)-algebra, Kolomogorov's 0-1 Law, lim sup and lim inf of a sequence of events, Fatou and reverse Fatou Lemma for sets, Borel-Cantelli Lemmas.
Integration and expectation, review and extension of elementary properties of the integral and convergence theorems [from Part A Integration for the Lebesgue measure on \(R\)]. Radon-Nikodym Theorem [without proof], Scheffé's Lemma. Integration on product space, Fubini/Tonelli Theorem. Different modes of convergence and their relations. Markov’s and Jensen’s inequalities. \(L^p\) spaces, Holder’s and Minkowski’s inequalities, completeness. Uniform integrability, Vitali’s convergence theorem.
Conditional expectation: definition, properties, uniquness. Conditional convergence theorems and inequalities, link with uniform integrability. Orthogonal projection in \(L^2\), existence of conditional expectation.
Filtrations and stopping times. Examples and properties. \(\sigma\)-algebra associated to a stopping time.
Martingales in discrete time: definition, examples, properties, discrete stochastic integrals. Doob’s decomposition theorem. Stopped martingales and Doob's Optional Sampling Theorem. Maximal and \(L^p\) Inequalities, Doob's Upcrossing Lemma and Martingale Convergence Theorem. Uniformly integrable martingales, convergence in \(L^1\). Backwards martingales and Kolmogorov's Strong Law of Large Numbers.