Difference between revisions of "Math 544: Advanced Probability 2"
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Revision as of 16:40, 3 April 2013
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.
 Convergence of random variables
 Almost sure
 In probability
 In mean
 In distribution
 Laws of Large Numbers
 Weak Law
 Strong Law
 Stochastic processes
 Gaussian
 Stationary
 Stationary increments
 Independent increments
 Filtrations
 Adapted processes
 Predictable processes
 Stopping times
 Ergodic theory
 Birkhoff's Ergodic Theorem
 Mixing
 Martingales
 Submartingales and supermartingales
 Doob Decomposition Theorem
 Optional Stopping Theorem
 Optional Sampling Theorem
 Martingale Convergence Theorem
 Convergence of backwards martingales
 Brownian motion
 Definition
 Existence
 Path properties
 Itô integral
 With respect to Brownian motion
 With respect to diffusion processes
 Itô 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