Syllabus
Math 595, CRN 53563, Section LD
Large Deviations
The study of atypical or rare events occupies
an important part in many probabilistic analyses. While there are many
types of rare events, the theory of exponentially small events
appears frequently enough that it deserves special consideration.
We will explore the circle of ideas known as large deviations,
which studies the structure and transformations of such exponentially
small events. We will attempt, when possible, to connect this theory
to financial problems, following some ideas of Pham; rare events are an
important part of the financial landscape, not only as cases of extreme risk,
but also as an integral part of investment-grade securities which, by
definition, should suffer losses only rarely.
There will be a lot of homework. I will ask you to prove a number of
elementary results and simplified calculations. The lectures will focus on
things which introduce what is new. The class runs until October 16.
Provisional Schedule
- Lecture 1: Some ``Large Deviations'' problems
- Lecture 2: Laplace asymptotics (and some convexity) (2 lectures)
- Lecture 3: Definitions and examples (1 lecture)
- Lecture 4: Contraction Principle and Varadhan's Lemma (1 lecture)
- Lecture 5: Gartner-Ellis and Cramer's theorem (2 lectures)
- Lecture 6: Sanov's theorem (1 lecture)
- Lecture 7: Schilder's Theorem (1 lecture)
- Lecture 8: Freidlin-Wentzell estimates (1 lecture)
- Lecture 9: Exit Problems (1 lecture)
- Lecture 10: Donsker-Varadhan results (2 lecture)
- Lecture 11: Losses in large pools (1 lecture)
- Lecture 12: Out-of-the-money-options (1 lecture)
- Lecture 13: Diversification and basket options (1 lecture)
- Lecture 14: Refined Asymptotics (1 lecture)
- Lecture 15: Importance Sampling (1 lecture)
References:
- Dembo and Zeitouni, Large Deviations Techniques and Applications, 2nd ed., 1996, Springer-Verlag
- Freidlin and Wentzell, Random Perturbations of Dynamical Systems, 2nd ed., 1989, Springer-Verlag
Grading: Grades will
be determined on the basis of homework (30%), a midterm (35%) and a
final (35%). I will attempt announce relevant things on my
Twitter Feed.
Richard Sowers
(Home Page)
Department of Mathematics
University of Illinois
at Urbana-Champaign
1409 W Green St.
Urbana, IL 61801
r-sowers@math.uiuc.edu
(217) 333-6246