Extended description: Foundation of probability theory, monotone classes and pi-lambda theorem, Kolmogorov extension theorem and infinite product spaces, Kolmogorov zero-one law, a.s. convergence, convergence in probability and in L^p of random variables, Borell-Cantelli lemma. Convergence of series of independent random variables: the theorems of Kolmogorov and Levy. Weak convergence of probability measures: characteristic functions, Levy-Cramer continuity theorem, tightness and Prohorov's theorem. The Central Limit Theorem: the Lindeberg-Feller theorem, the Levy-Khintchine formula, stable laws. Conditional expectation. Discrete time (sub- and super) martingales: Doob's maximal inequality, Optional Stopping Theorem, uniform integrability, and the a.s. convergence theorem for L¹ bounded martingales, convergence in L^p. Definition, existence and basic properties of the Brownian Motion. Other topics in probability theory at the choice of the instructor (e.g. Markov chains, Birkhoff-Khinchine and Kigman ergodic theorems, Levy's arcsine law, Law of Iterated Logarithm, convergence to stable laws). See also Probability Prelim Study Guide. Textbook choices: Real Analysis and Probability by R.M. Dudley. Probability, Theory and Examples, 4th edition by R. Durrett. Probability Theory by S.R.S. Varadhan. Foundations of Modern Probability by O. Kallenberg. Supplementary reading: Lecture Notes by R. Bass. A Course in Probability by K.L. Chung. Probability with Martingales by D. Williams.
Prerequisites: MATH 5111
Credits: 3

Extended description: The course material changes with each occurrence of the course and may be taken for credit repeatedly with the instructor's permission. Continuous time random processes, Kolmogorov's continuity theorem. Brownian Motion: the Donsker invariance principle, Holder continuity, quadratic variation. Continuous-time martingales and square integrable martingales. Markov processes and the strong Markov property. Properties of Brownian Motion: strong Markov property, Blumenthal zero-one law, Law of Iterated Logarithm. Stochastic integration with respect to continuous local martingales, Ito's formula, Levy's Characterization of Brownian motion, Girsanov transformation, Stochastic Differential Equations with Lipschitz Coefficients. Other topics in probability theory and stochastic processes at the choice of the instructor (e.g. connection to PDEs, local time, Skorokhod's embedding theorem, zeros of the BM, empirical processes, concentration inequalities and applications in non-parametric statistics, infinite dimensional analysis, processes with stationary independent increments and infinitely divisible processes, jump measures and Levy measures, random orthogonal measures, symmetric Markov processes, stochastic analysis for jump process). Textbook choices: Foundations of Modern Probability by O. Kallenberg. Brownian Motion and Stochastic Calculus, 2nd Ed. by I. Karatzas and S. Shreve. Continuous Martingales and Brownian Motion, 3rd Ed. by D. Revuz and M. Yor. Supplementary reading: Lecture Notes by R. Bass. Stochastic Processes by S.R.S. Varadhan. Introduction to the Theory of Random Processes by N.V. Krylov.
Prerequisites: MATH 5160
Credits: 3