Rohini Kumar (Instructor, first part)
Office: South Hall 5505
kumar at pstat.ucsb.edu

Dates: 1/13, 1/20, 1/27, 2/3 [Syllabus] Homework(TBA)

The study of probabilities of very rare events falls under the rubric of Large deviations. In this introductory course we will cover some basic definitions in large deviation theory and techniques used in obtaining large deviation results. For the most part, we will restrict ourselves to the finite dimensional setting. Some applications in Finance will be discussed.

Topics:

1. Introduction (1/13)
• 1.1 Motivation
• 1.2 Defintions
• 1.3 Large Deviation Principle (LDP)
2. Key Theorems (1/20, 1/27)
• 2.1 Cramer's Theorem
• 2.3 Sanov's Theorem
• 2.4 Gartner Ellis Theorem
3. Applications to Option Pricing (2/3)
• 3.1 Short maturity, Out-of-the-Money Call Option pricing

Textbook: "Large Deviations Techniques and Applications" by Amir Dembo and Ofer Zeitouni

Tomoyuki Ichiba (Instructor, second part)
Office: South Hall 5514
ichiba at pstat.ucsb.edu

Dates: 2/10, 2/17, 2/24, 3/3 [Syllabus] Homework(TBA)

In this second part we cover the introductory theory of superprocesses that are measure-valued stochastic processes. First we study branching Brownian motions and supper Brownian motions via martingale problems. Then we study more general classes of superprocesses. The study of such processes is mostly applied to Mathematical Population Genetics but we explore some possible applications to Mathematical Finance. [More]