MATHEMATICS 565B
Continuous Time Stochastic Processes

FALL 2007

Overview. For this semester, we will develop ideas in continuous time stochastic processes. We begin by developing basic tools in stochastic processes - filtrations and stopping times and notions of equivalence of stochastic processes. We will then study Levy processes as the continuous time version of random walks. The principle classes of processes that we will investigate are martingales and Markov processes and the connections between them.

This will lead us to a classical analysis of one dimenstional diffusions and a more modern analysis using stochastic integrals.

We will use no one particular text. I will provide continually updated class notes. Some of my sources for material are

• Probability by Leo Brieman
• Probability: Theory and Examples by Richard Durrett
• Markov Processes: Characterization and Convergence by Stewart N. Ethier and Thomas G. Kurtz
• An Introduction to Probability Theory and Its Applications, Volumes I and II by William Feller
• A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray
• Stochastic Flows and Stochastic Differential Equationsby Hiroshi Kunita
• Continuous Martingales and Brownian Motion by Daniel Revuz and Marc Yor
• Probability with Martingales by David Williams
• Diffusions, Markov Processes, and Martingales, Volume I by L. C. G. Rogers and David Williams
• Diffusions, Markov Processes and Martingales,Volume II Ito Calculus by L. C. G. Rogers and David Williams

Day to Day Operations. The course meets for lecture each Tuesday and Thursday from 9:30 to 10:45 in the Mathematics Building Room 501.

If you need to contact me, write me electronically at jwatkins@math.arizona.edu, call me at 621-5245, or drop by my office Math 520. My office hours will be set in the next couple of days.

Topics. An approximate course outline is as follows:

• Basic ideas on stochastic processes in continuous time
  • Notions of equivalence
  • Sample path properties
  • Filtrations
  • Stopping times
• Levy Processes
• Martingales
  • Sample path regularity
  • Maximal inequalities
  • Law of the iterated logarithm for Brownian motion
• Markov processes
  • Transition operators
  • Operator semigroups
  • Hille-Yosida theorem
  • Strong Markov property
  • Connections to martingales
  • Jump processes
  • Sample path regularity for Feller processes
  • Transformations of Markov processses
  • Stationary measures
  • One dimensional diffusions
• Stochastic integrals
  • Ito integral
  • Ito formula
  • Stochastic differential equations
  • Applications to diffusions

Evaluation of Students. We will evaluate your work in the course through your performance on homework assignments and an end of term project..

If you fail to complete the course due to circumstances unforeseen, then you may qualify for a grade of I, ``incomplete'' if all of the conditions are met

• You have completed all but a small portion of the required work.
• You have scored at least 50% on all work completed.
• You have a valid reason for not completing the course on time.
• You agree to make up the material in a short period of time.
• You ask for the incomplete before grades are due - 48 hours after the final exam.

Best wishes to you for a good semester in this course and in all your other activities.

Joe Watkins

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