Course Name: Stochastic Processes

Course Code: S2293026

Semester: 1

Credit: 2

Program: Electronics

Course Module: Compulsory

Responsible: Xue Yang

E-mail:yang2013@tju.edu.cn

Department:School of Mathematics, Tianjin University

Time Allocation(1 credit hour = 45 minutes)

Course Description

The course covers the following topics:

• One-Parameter Processes, Usually Functions of Time

• Markov Processes

• Stochastic Calculus, Diffusions, and Spectra

• Ergodic Theory

• Large Deviations Theory

Prerequisite

Mathematics, Probability, Mathematical Reasoning.

Course Objectives

This is a course on basic stochastic processes and applications with an emphasis on problem solving. Topics will include discrete-time Markov chains, Poisson point processes, continuous-time Markov chains, and renewal processes. The material will be treated in a mathematically precise fashion although some proofs will be skipped due to time limitations.

Course Syllabus

1. Basics

   1. Definition of stochastic processes, examples, random functions

   2. Finite-dimensional distributions (FDDs) of a process, consistency of a family of FDDs

   3. Theorems of Daniell and Kolmogorov

   4. Probability kernels and regular conditional probabilities

2. One-Parameter Processes, Usually Functions of Time

   1. Examples of one-parameter processes

   2. Shift-operator representations of one-parameter processes

   3. Three kinds of stationary, the relationship between strong stationary and measure-preserving transformations

   4. Kac's Recurrence Theorem

3. Markov Processes

   1. Markov processes and their transition-probability semi-groups

   2. Markov processes as transformations of IID noise

   3. Markov processes as operators on function spaces

   4. Generators of homogeneous Markov processes, analogy with exponential functions

   5. The strong Markov property and the martingale problem

4. Stochastic Calculus, Diffusions, and Spectra

   1. Diffusions

   2. Wiener measure; non-differentiability of almost all continuous curves

   3. Heuristic approach to stochastic integrals (via Euler's method)

   4. Rigorous approach to stochastic integrals

   5. Physical Brownian motion

5. Ergodic Theory

   1. Introduction to ergodic properties and invariance

   2. Metric transitivity

   3. Ergodic decompositions

   4. Central limit theorem for strongly mixing sequences

6. Large Deviations Theory

   1. Large deviations principles and rate functions

   2. Cramer's theorem on large deviations of empirical means

   3. Large deviations for Markov sequences

Textbooks & References

• Durrett.Essentials of Stochastic Processes. Springer(2nd ed).

We may also use readings from a few textbooks:

• Greg Lawler.Introduction to Stochastic Processes. Chapman and Hall.

• Sheldon Ross.Stochastic Processes. Wiley.

Grade Distribution

Attendance: 10% Homeworek:30% Final Exam:60%

Capability Tasks

CT1: To understand basic science, and to have analytical ability and the ability to integrate related knowledge.

CT2: To apply relevant professional knowledge to the field of science and technology: understanding of the basic concepts and its connotation, application of different methods and concepts which have been learned, capability of judging the scope and limitations of such applications.

CT3: To grasp methodologies and engineering tools: identifying, utilizing and solving problems. Even if the students are not familiar with the content, they can turn to computer tools for systematic analysis.

CT4: To carry out experiments in research environment with the abilities to utilize tools, especially for data collection and processing.

CT10: To have the capacity to work in international environment; the capability to master one or more foreign languages and be open to foreign cultures; be able to acclimatize themselves to the international language environment.

Achievements

• To be able to recognize the major characteristics of a random signal. - Level: M.

• To make the link between correlation and energy. - Level: M.

• To be able to model a random signal by an AR process. - Level: A.

Students: Electronics, Year 1