Syllabus

• Instructor: Richard Sowers
• Office: 347 Illini Hall
• Phone: (217) 333-6246
• email: r-sowers@illinois.edu
• Home page: https://math.uiuc.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