Large Deviations

**Instructor:**Richard Sowers**Office:**347 Illini Hall**Phone:**(217) 333-6246**email:**r-sowers@illinois.edu**Home page:**`https://faculty.math.illinois.edu/~r-sowers`(this syllabus can be found there)**Office Hours:**10:00-11:00 MW and by appointment**Class meets:**MWF 11-11:50 A.M. in 345 Altgeld**Text:**My Notes

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.

- 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)

- 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

Department of Mathematics

University of Illinois
at Urbana-Champaign

1409 W Green St.

Urbana, IL 61801

r-sowers@math.uiuc.edu
(217) 333-6246