Advanced Stochastic Processes

MATH8007, Fall semester, 2017

Instructor: Yizao Wang
Email: yizao.wang@uc.edu
Office: 4302 French Hall
Office Hours: By appointment.
Class meeting: MW 2:30-3:50pm, Room 115 WCharlton.

Course description

The course will cover a series of classical stochastic models. The majority models are representative ones of the so-called combinatorial stochastic processes, with motivations/applications from population genetics and non-parametric inference, among others. Discrete-time martingales and Poisson point processes, two classes of fundamental stochastic processes in modern probabilty theory, will be introduced gradually, with an emphasis on their role in the analysis of the models of interest.

Students will have reading assignments on either related models that continue the investigation in class, or those of students' own interest. Stochastic models that will be presented by the instructor or may be selected by students for independent/group studies will come from the following areas

• Statistics
• Population genetics
• Machine learning
• Finance, insurance, and risk management
• Mathematical physics
• Operations research
• Potential theory

Pre-req: Applied probability (MATH 6008) or equivalent would be helpful.

Grades

Oral presentations (40%), a written report (40%) based on reading assignments, and classroom participation (20%). No exams/homeworks.

Tentative Schedule

• Week 1 (Aug 21): Stochastic processes, an overview.
• Week 2 (Aug 28): Random walks.
• Week 3 (Sep 4): Labor Day, no class on Mon. Martingales.
• Week 4 (Sep 11): Martingales.
• Week 5 (Sep 18): Polya's urn model.
• Week 6 (Sep 25): Introduction to random graphs.
• Week 7 (Oct 2): Branching processes.
• Week 8 (Oct 9): Reading Day, no class on Mon. Moran model and Wright-Fisher model, geneology and coalescent.
• Week 9 (Oct 16): Continuous-time Markov chain.
• Week 10 (Oct 23): Kingman's coalescent.
• Week 11 (Oct 30): Chinese Restaurant process.
• Week 12 (Nov 6): Exchangeable random partitions.
• Week 13 (Nov 13): Dirichlet process, Gibbs sampling.
• Week 14 (Nov 20): Dirichlet process and Dirichlet process mixture model.
• Week 15 (Nov 27): No class on Mon, last class on Fri, same time/room. Coalescent trees.

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