Math 544: Advanced Probability 2

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Catalog Information


Advanced Probability 2.

Credit Hours



Math 543.


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 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.

  1. Convergence of random variables
    • Almost sure
    • In probability
    • In mean
    • In distribution
  2. Laws of Large Numbers
    • Weak Law
    • Strong Law
  3. Stochastic processes
    • Gaussian
    • Stationary
    • Stationary increments
    • Independent increments
    • Filtrations
      • Adapted processes
      • Predictable processes
      • Stopping times
  4. Ergodic theory
    • Birkhoff's Ergodic Theorem
    • Mixing
  5. Martingales
    • Submartingales and supermartingales
    • Doob Decomposition Theorem
    • Optional Stopping Theorem
    • Optional Sampling Theorem
    • Martingale Convergence Theorem
    • Convergence of backwards martingales
  6. Brownian motion
    • Definition
    • Existence
    • Path properties
  7. Itô integral
    • With respect to Brownian motion
    • With respect to diffusion processes
    • Itô 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