Math 546: Continuous Time Stochastic Processes

Course Information

• Instructor: Omer Angel, angel@math.ubc.ca
• Lectures: MWF 13:00-14:00 at MATH 203
• Office hours: TBA at Math Annex 1210
• Prerequisites: Math 545 or consent of the instructor

Textbook and References

Primary textbook:

Diffusions, Markov Processes and Martingales, vol. 2: Itô Calculus, by Rogers and Williams, Cambridge University Press, 2000. (Note: The text occasionally refers to Volume 1).

Other References:

• [B] Breiman, Probability (Good basic reference for measure-theoretic probability)
• [D] Durrett, Probability: Theory and Examples (Good basic reference for measure-theoretic probability)
• [EK] Ethier and Kurtz, Markov Processes: Characterization and Convergence
• [RY] Revuz and Yor, Continuous Martingales and Brownian Motion
• [P] Protter, Stochastic Integration and Differential Equations
• [W1] D. Williams, Probability with Martingales (Good reference for discrete parameter martingales)
• [W] D. Williams, Diffusions, Markov Processes and Martingales Vol. 1

Course Outline and Focus

This is a rigorous course on finite dimensional continuous Markov processes. Most topics covered will be included in Chapters IV and V of the Rogers and Williams text.

• The course will study stochastic integration with respect to continuous semimartingales, and Itô's stochastic calculus.
• The focus will be on finite-dimensional stochastic differential equations (SDEs).
• Topics include: Review of Brownian motion, Itô's pathwise uniqueness results, weak solutions, martingale problems, and the relationship with strong or pathwise solutions.
• Change of measure (Girsanov) formulae will be derived and applied to the well-posedness of the martingale problem for finite dimensional SDEs.
• Depending on time, the course may study local times and one-dimensional diffusion theory or Stroock-Varadhan martingale problems.

Assumed Background

• The course assumes familiarity with measure theoretic probability theory, including discrete parameter martingale theory and Brownian motion. (A brief review of the last two topics will be provided).
• Students from other Departments may treat measure theoretic prerequisites as "black boxes".

Evaluation

Evaluation will be based on homework assignments, which will be given every 2-3 weeks.