# Course Details:

Lecturer: Jeffrey Collamore; ph.: 3532 0782; e-mail: collamore-at-math.ku.dk.

Lectures: Monday 9-12 in Aud. 10; Tuesday 15-17 in Aud. 10.

Evaluation: There will be a 30-minute oral exam and exercises. Your grade will be based primarily on the oral exam, although the exercises will also count for a small part of the grade. In particular, it is required that all exercises are attempted and the exercise set is "passed" in order to participate in the oral exam.

Prerequisites: A basic measure-theoretic course in probability theory.

Course material: Lecture notes.

Course description: This will be an introductory course in the theory of large deviations and its applications. Topics will include: Cramer's theorem for sample means and for level crossings, moderate deviations, heavy tails, and a brief introduction to the multidimensional theory. Applications will include insurance mathematics and finance, statistics, and Monte Carlo methods (importance sampling).

# Schedule for the lectures:

03.09.18: Introduction (JC pp. 4-6); Cramer's theorem in R (JC, pp. 7-10).
04.09.18: Cramer's theorem in R (JC, pp. 7-10, cont.); proof of lower bound (JC, pp. 10-12). Note: The truncation and smoothing techniques are described in the notes starting on p. 12; this is needed for a complete proof of the lower bound.
10.09.18: Proof of upper bound (JC, pp. 14-16); sharp asymptotics (first handout, pp. 16-20).
11.09.18: The one-dimensional ruin problem via large deviation methods (preparation for the multidimensional ruin problem, see below).
17.09.18: Cramer's theorem in d-dimensional Euclidean space (JC, pp. 27-36).
18.09.18: Cramer's theorem in higher dimensions, cont. (JC, pp. 27-36).
24.09.18: The multidimensional ruin problem (JC, pp. 51-59).
25.09.18: Multidimensional ruin, cont.; large deviations for paths (JC, pp. 44-50).
01.10.18: LDs for paths cont.; Sanov's theorem (second handout).
02.10.18: Gibbs conditioning; importance sampling (JC, pp. 63-73).
08.10.18: Importance sampling cont.; moderate deviations and heavy tails.
09.10.18: Concentration inequalities (third handout). Last lecture.

"JC" refers to my lecture notes posted on Absalon, while the "handouts" refer to the book excerpts given out in the lectures.

# Homework:

One mandatory homework set has been posted on Absalon (due October 2). A second homework set will be posted around the fourth week of the course (approximately September 28), while a short third homework set will be posted around the sixth week of the course (approximately October 9).