Math 637: Advanced Probability 2

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Catalog Information


Advanced Probability 2.

Credit Hours



Math 636.


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


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


This course has Math 636 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 637 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.

  1. The Daniell-Kolmogorov Theorem
  2. Stochastic processes and filtrations
  3. Continuous-time Martingales
  4. Brownian motion
  5. Gaussian processes
  6. Levy processes
  7. Regular conditional probabilities
  8. Markov processes
  9. Stochastic integration
  10. Ito’s formula


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