with L a Levy process (LP) noise (i.e., independent stationary
increment), or similar (see below). In
traditional Brownian motion (BM) stochastic calculus, L is continuous,
but in this course, using so called general theory, it may have jumps.
(The main difficulties with SDE are probabilistic, and quite
little ordinary differential equation techniques come into
play.) The applied part of the course focus on application of such SDE to
model the evolution in time of real-world phenomena.
[Specifically, in traditional stochastic calculus, L and
the integral int_0^t ... dL are
continuous local martingales, i.e., continuous
time-changed BM's,
while in general theory, L and the integral are semimartingales,
i.e., "conditional" LP (which are close to but also considerably more
general than LP). Most results in general theory
can be derived for LP (which are
intuitive). Then the result will also hold, and look exactly the same, for (less intiutive) general semimartingales, when
reinterpreted for "conditional"
LP. It is the intention to stress this in the course.]
The "general theory" is the most general framework to which
traditional stochastic calculus extends, keeping in essence much of
the ideas from the traditional approach (suitably modified
for jumps). Nevertheless, the resulting theory
is much more general than the traditional. It is
required in important contemporary applications and modelling, where continuous noise is
insufficient. If one jumps some very difficult proofs (which there is little reason to do anyway), the general
theory is not much more difficult to take in than is the traditional.
The main theoretical content of the course (5 credits, or so) will be the general
theory with Levy processes. Clearly, many proofs have to be omitted (a few of which one
can spend whole courses doing). There is nothing
wrong with this, and there will be more than enough proofs left to do
anyway.
Traditional stochastic calculus is covered by specializing to noises
without jumps, as is basic theory for continuous time martingales.
The course will have an important applied part (2 credits, or so), where individually chosen
applied articles are studied, individually or in small groups. Naturally,
the articles should use Levy processes (/general theory) in
modelling. This will give (much) improved understanding of the theory, as well as of modelling practices in a specific area. For most
participants the articles will be on mathematical finance, but
for those who lack such
interests, we will find something else. (For
mathematicians there are articles e.g., on Levy differential geometry and
Levy generator symbolic analysis.) Such articles are often less difficult to
read than one might initially expect. The
plan is to summarize the articles in "junior seminars", that will make out the examination procedure of the
course.
At minimum, the course should give participants a working ability
to take in applied literature using Levy process (/general theory)
modelling.
[While all this may look cute, it is a fact
that a complete understanding of the general theory is difficult, so
there will for sure be
things to do also for "senior" participants.] [It is quite suitable to "move into" this
area at this time. Undergraduate master thesises may deal with one of many loose
ends that can be found in applied
articles. Graduate students may look at a gemeralization of traditional modelling in one application (applied area), and then work on adapting it to another, starting out
quite applied, but with a natural entrance to a possibly more
theoretical focus.]
Timeframe: The plan is to schedule lectures September-October, and agree on a time, preferably common for
all participants, before the end of
November say, to complete the applied part.
Prerequisites: Graduate students
should have taken a graduate
course in probability. (Graduate students with reasonable knowledge of
"pure" graduate mathematics will do well with undergraduate probability.) Senior
undergraduates with a background in mathematical finance can take the course if specially prepared in "soft" traditional BM
stochastic calculus. Graduate
students do not need such preparation, but will gain deepened
understanding from it. The books "Klebaner:
Introduction to Stochastic Calculus with Application" (quite soft) and
"Mikosch: Elementary Stochastic Calculus" (very soft) are fine.
Literature: See the above announcement for indications
of literature.
Responsible for the course is Patrik Albin. Questions are wellcomed.
