### MATH 5161: Probability Theory and Stochastic Processes II

**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). With a change of content, this course is repeatable to a maximum of six credits.

**Prerequisites:** MATH 5160.

**Credits:** 3

**Sections: **Fall 2018 on Storrs Campus