Home Page: Topics in Large Deviations and
Finance; Block 1,
2011
Course Details:
Lecturer:Jeffrey
Collamore; ph.: 3532 0782; e-mail: collamore-at-math.ku.dk. Lectures: Monday 10-12 in Aud. 9; Tuesday 13-15 in Aud. 8
(Weeks 36-41 and 43).
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 and articles, to be posted on Absalon.
Course description:
This will be an introductory course in large deviations and its
applications to insurance and finance. We will begin by describing
the most fundamental results (e.g., Cramer's theorem, the
Gartner-Ellis theorem, the multidimensional ruin
problem, large deviations for sample paths, etc.). The emphasis will be on understanding these results and
their applications, i.e., we will not prove the more technical
results in any detail (although proofs of a some basic results,
such as Cramer's theorem, will be given). In the second part of
the course, we will look at a couple recent articles, where large
deviations methods have been applied to problems mainly in the
area of financial mathematics.
Schedule for the lectures:
05.09.11: Introduction. (Lecture notes: Ch. 1.)
06.09.11: Cramer's theorem: statement and examples. (Lecture notes: Ch. 2,
pp. 6-9.)
12.09.11: Cramer's theorem: proof. (Lecture notes: remainder of Ch. 2, except you can omit discussion on "truncation" and "smoothing" in Lemmas 2.3 and 2.4.)
13.09.11: Cramer's theorem: proof, cont.
19.09.11: Cramer's theorem in d-dimensions with proof (Ch. 4, again omitting
the discussion on truncation and smoothing in the proof of the lower bound.
Pay particular attention to the proof of the upper bound, which involves
new ideas.)
20.09.11: Cramer's theorem in d-dimensions with proof, cont. (Ch.4); statement
of the Gartner-Ellis theorem (Ch. 3, pp. 16-20).
26.09.11: The multidimensional ruin problem (Ch. 8 through p. 52).
27.09.11: The multidimensional ruin problem, cont.
03.10.11: Description of LDs for sample paths (Ch. 7 through p. 40);
importance sampling (Ch. 9). Note:
This lecture will take
place 8:45-11:30 in Aud. 9.
04.10.11: Cancelled.
10.10.11: Importance sampling, cont. Note:
This lecture will take place 9:15-12:00 in Aud. 9.
11.10.11: Original articles: Portfolio credit risk (Dembo et al.).
24.10.11: Original articles: Portfolio credit risk, cont.;
portfolio investment theory (Stutzer).
25.10.11: Original articles: Portfolio investment theory.
Exam date: November 11 in Aud. 5, 9:00-15:30.
(Censors: Magdalena Musat and Mogens Steffensen.) Reexam date: February 3.
Literature:
Collamore, J.F. Lecture notes on large deviations. Ch. 1-4, 7-9.
Dembo, A., Deuschel, J-D., Duffie, D. (2004). Large portfolio losses.
Finance Stochast. 8, 3-16.
Stutzer, M. (2003). Portfolio choice with endogenous utility:
a large deviations approach. J. Econometrics 116, 365-386.
Suggestions about the reading. Lecture notes: Ch. 1: This is just an introduction (read all). Ch. 2: In the proof of the lower bound, you can ignore the part on
truncation and smoothing, i.e., read only pp. 6-11. Then read the
proof of the upper bound on pp. 13-15. Ch. 3: You are only responsible for the statement of the
theorem (pp. 16-20, say). You do not need to read the proof
(although you might want to observe that it is essentially the
same as the proof of Cramer's theorem in Ch. 2).
Ch. 4: As with Ch. 2, read everything except the truncation
and smoothing used in the proof of the lower bound; i.e., you can skip
pp. 30-31 (and pp. 34-36 are optional). Pay particular attention to
the proof of the upper bound, which involves some new ideas.
Ch. 7: Concentrate on the statement of the theorem and
the examples. You can skip the discussion of the formal proof
given on pp. 41-43.
Ch. 8: Read the entire proof (including the extension, in
the upper bound, from compact sets to general closed sets). You may omit the discussion of extensions
on pp. 52-55.
Ch. 9: The discussion of Siegmund's algorithm on pp. 58-60
is optional, as is the proof of "necessity" on pp. 63-64. Briefly read the
short discussion on IS for sample means, even though we didn't cover
this in lecture.
Article 1 (Dembo et al): Concentrate on the first part
of the article, namely, pp. 3-10.
Article 2 (Stutzer): Concentrate on pp. 365-379.
Homework:
Homeworks 1, 2, and 3 have all been posted on Absalon (and there
will be no further homework sets beyond these three sets).
All homework problems should be handed
in preferably by the time of the last lecture or by Friday,
October 27 (either to me
personally or to my mailbox on the
first floor-see the secretaries in Room 1.03). (Note: For the last
homework problem, the lecture on Monday, October 24 will be relevant.)
Exam topics:
1. Cramer's theorem in one dimension.
2. Cramer's theorem in higher dimensions: upper bound (and extension
for Gartner-Ellis sequences).
3. Cramer's theorem in higher dimensions: lower bound (and extension
for Gartner-Ellis sequences).
4. Multidimensional ruin problem (and rough statement of Mogulskii's
theorem and their connections).
5. Importance sampling.
6. Credit risk application (article of Dembo et al).
7. Portfolio theory application (article of Stutzer).