# Math 637

## Contents

## Catalog Information

### Title

Advanced Probability 2.

### Credit Hours

3

### Prerequisite

### Recommended

Math 341, 342, Stat 441(?); or equivalents.

### Description

Advanced concepts in modern probability. Convergence theorems and laws of large numbers. Stationary processes and ergodic theorems. Martingales. Diffusion processes and stochastic integration.

## Desired Learning Outcomes

### Prerequisites

This course has Math 543 as a prerequisite, so it can build on the work done in that class.

### Minimal learning outcomes

Outlined below are topics that all successful Math 544 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

- The Daniell-Kolmogorov Theorem
- Stochastic processes and filtrations
- Continuous-time Martingales
- Brownian motion
- Gaussian processes
- Levy processes
- Regular conditional probabilities
- Markov processes
- Stochastic integration
- Itô's formula

### Textbooks

Possible textbooks for this course include (but are not limited to):

- Achim Klenke,
*Probability Theory: A Comprehensive Course*, Springer, 2008.

### Additional topics

### Courses for which this course is prerequisite

None