Department of Mathematics
Uppsala University

These lectures are directed to a general audience of PhD students.
The series is named after Professor Matts Essen (1932-2003), who worked at Uppsala until his retirement 1997

June 11-15, 2018: Ioannis Karatzas (Columbia)

Stochastic Finance

We will develop a mathematical theory for finance based on the principle that it should not be possible
to fund, starting with arbitrarily small initial capital, a cumulative withdrawal stream which is
nontrivial, that is, strictly positive with positive probability. In the context of a market with
continuous asset prices modeled by semimartingales, we shall show that proscribing such egregious forms
of arbitrage (but allowing for the possibility that one portfolio might outperform another) turns out to
be equivalent to any one of the following conditions:

(i) a portfolio with the local martingale numeraire property exists;
(ii) a growth-optimal portfolio exists;
(iii) a portfolio with the log-optimality property exists;
(iv) a strictly positive local martingale deflator exists; and
(v) the market has locally finite maximal growth.

We will give precise meaning to all these terms, and show that the above five equivalent conditions are
descriptive - in that they can be formulated entirely, in fact very simply, in terms of the local
characteristics of the underlying asset prices. Full-fledged theories for hedging and for
portfolio/consumption optimization can be developed in such a setting, as can the important notion of
"market completeness".

The lectures will start by reviewing the necessary background from stochastic analysis, to ensure they
are accessible to all graduate students in mathematics; no prior knowledge of finance will be needed or
assumed. The draft of a graduate text with working title "Arbitrage Theory via Numeraires" (joint work
with Constantinos Kardaras) will be made available to participants, who will be asked to complete
assignments drawn from it.

Schedule (June 11-15, 2018):

10.15 - 11.00  Lecture
11.15 - 12.00  Lecture
13.30 - 14.30  Lecture/Problem Session

Venue: Room 4006, Aangstroemlaboratoriet, Uppsala

There will be one written assignment. The course gives 3 ECTS credits.

Event's Dinner (provisionally):  Friday 17.00 at Peppar Peppar

Lunches (provisionally):  Monday-Friday 12.20 at Eklundshof

Registered Participants:

Alexander Aurell (KTH)
Erik Ekstroem (Uppsala)
Ioannis Karatzas (Columbia)
Dan Lilja (Uppsala)
Volodymyr Mazorchuk (Uppsala)
Andrea Pasquali (Uppsala)
Konstantinos Tsougkas (Uppsala)
Jakob Zimmermann (Uppsala)

Previous lectures:

2017 Jacob Rasmussen (Cambridge)  Knot polynomials and knot homologies
2016 Christian Krattenthaler (Vienna)  The art of bijection
2015 Wilhelm Schlag (Chicago)  An introduction to nonlinear wave equation
2014 Francois Baccelli (Austin)  Stochastic geometry
2013 Alexander Kleshchev (Oregon)  The symmetric group: its combinatorics, representations and applications
2012 Jens Marklof (Bristol)  Emerging applications of homogeneous dynamics: From discrete mathematics to statistical physics