M 394C Stochastic Processes II 

(meets with 68795 CSE 394)

 

Mihai Sirbu

TTH 12:30 – 2:00

RLM 11.176

 

Course description: (Topics in) Stochastic Analysis consists of a selection of related topics roughly following after Theory of Probability I and II, and is independent of other Topics courses in the field.
The emphasis is on Stochastic Differential Equations (SDE's) and optimization problems involving such equations (stochastic control problems). Actually, most of the time will be dedicated to a special class
of optimization problems called Optimal Stopping. For Optimal Stopping Problems, we can present a complete and rigorous probabilistic treatment (with only some comments on the corresponding analytic theory).
A significant amount of time will be spent on (mostly explicit) examples of Optimal Stopping. Some (actually most) such examples are related to Finance, but the emphasis will be on the mathematical part.
The course assumes some understanding of discrete-time probability (Theory of Probability I) and of the Stochastic Integration Theory (Theory of Probability II). 

Topics covered by the course:

  •    Stochastic Differential Equations (SDE) and Markov property of SDE's
  •    Optimal Stopping Problems (in Markov and non-markov cases) and applications. Most applications are from Finance, but no knowledge of Finance is required. This is intended as the largest part of the lecture.
  •    Other classes of stochastic control problems, beyond optimal stopping (time permitting)


Intended audience:

  • graduate  students in mathematics with an interest in probability, stochastic processes, financial mathematics and applied mathematics in general, as well as
  • graduate students in natural sciences, engineering, economics, business, or any other field where Stochastic Differential Equations,  Continuous-Time stochastic Processes and Stochastic Control are  used


Prerequisites: it is  recommended, but not required,  that students have taken Theory of Probability I and II (M385C and M385D). However, there is no hard prerequisite and students who have knowledge of
 stochastic integration and Ito's Formula should be fine. Please ask  the instructor any question you may have about the background needed.

Textbook/Literature:  there is no particular book that is required, as the lectures attempt to be reasonably self-contained.  However, the lecture will be based on material drawn from the following set of books:

  • I. Karatzas and S.E. Shreve: Brownian Motion and Stochastic Calculus, Second Edition, Springer (for  SDE's)
  • D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer (SDE's)
  • Shiryaev: Optimal Stopping Rules
  • Peskir and Shiryaev: Optimal Stopping and Free Boundary Problems
  • Stroock and Varadhan: Multidimensional Diffusion Processes