Download STATS 620: Applied Probability and Stochastic Modeling Winter 2016

STATS 620: Applied Probability and Stochastic Modeling
Winter 2016
Summary: This course offers an introduction to stochastic processes through models, techniques,
and applications. It starts with a review of probability theory and then provides a high-level
overview of stochastic process theory. Introduced are fundamental models such as random walks and
Brownian motion; branching processes and Markov chains; Poisson, renewal and related processes;
Continuous time Markov Chains, Markov and diffusion processes. The course will strive to develop
3 type of skills: (i) general understanding of fundamental theory and models (ii) technical skills and
intuition via pencil and paper problem-solving (iii) intuition through computer simulation, Monte
Carlo methods, and applications.
Formal Prerequisites: MATH 451 or equivalent knowledge of real analysis. Knowledge of probability at the level of BIOSTAT 601 or MATH 525.
Required Text: Adventures in Stochastic Processes by Sidney I. Resnick.
Recommended Text: Stochastic Calculus and Financial Applications by Michael J. Steele.
Course details:
◦ Lectures: Tue & Thur, 11:30 am – 1:00 pm in 1512 CC Little.
◦ Instructor: Stilian Stoev, [email protected].
◦ Office hours: Monday and Thursday, 3:15 pm – 4:45 pm in 445C West Hall, or by appointment.
◦ GSI grader: Can Le, [email protected]; Office hours: Friday, 2:00-3:30 pm in 438 West Hall.
◦ Credits: provides 3 credits.
◦ Website: Canvas.
Grading and homework: Homework will be assigned about every 1 or 2 weeks. It is planned
to be an important part of your learning experience. All solutions to the homework and exam
problems you submit should be your own work. You are required to comply with the University of
Michigan and Rackham Graduate School academic honor code.
http://www.rackham.umich.edu/policies/academic and professional integrity/
No late homework will be accepted, unless an extension is granted explicitly by the instructor
upon request by the student ahead of time. The final grade will be based on the homework (30%),
midterm (35%) and final (35%) exams.
Exam Schedule:
• Midterm: Thursday, February 25, 11:30 am – 1 pm (in class, 1512 CC Little)
• Final: Thursday, April 21: 1:30 pm - 3:30 pm (in 1512 CC Little).
General advise: Attempt all homework problems as soon as you receive them. You are encouraged
to discuss problems and concepts with the instructor, fellow students, and the GSI, but do so after
you have thought about them on your own. Please do not hesitate to go to the GSI’s or instructor’s
office hours. The homework and exam solutions you submit should, however, be your own work.
Tentative Schedule and List of Topics
• (1 weeks) Review of Probability Theory I: probability spaces and σ-fields; random variables
and distributions; independence and conditioning; expectation, limit theorems, modes of
convergence and LLN; conditional expectation – an L2 -perspective;
• (1 week) Review of Probability Theory II: Popular discrete and continuous models; Joint distributions and transformations; Order statistics; Weak convergence and characteristic functions; the CLT and infinitely divisible distributions. The Multivariate Normal distribution.
• (1 week) Stochastic Processes – an overview: existence (Kolmogorov) – the fdd and random element perspective. Examples: time series, Brownian motion, Poisson process, Gaussian processes, Markov processes. Versions, separability, measurability, and path properties
(Kolmogorov-Chentzov). Weak convergence of stochastic processes.
• (1 week) The Simple Branching process, the Simple Random walk, Stopping times and the
Wold’s identity. (Ch. 1)
• (2 weeks) Markov Chains (Ch. 2)
• (1 week) Renewal Theory (Ch. 3)
• (1.5 weeks) Point processes (Ch. 4)
• (1.5 weeks) Continuous Time Markov Chains (Ch. 5)
• (2 weeks) Martingales: inequalities and convergence; optional sampling; Applications to gambling and random walks. Kakutani’s dichotomy and likelihood ratios;
• (1 week) The Brownian motion (Ch. 6)
• (1.5 weeks) Diffusion processes and Weiner-Itô stochastic integrals.