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

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)

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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