From 14.10.2013 to 15.02.2014

Monday: 10-12 MA-651 Mathematics building Strasse der 17 Juni

Tuesday: 10-12 MA-744 Mathematics building Strasse der 17 Juni

Risk and uncertainty are two concepts which became — and not only as a consequence of the recent economical crisis — more and more important in public perception and discourse. Though intuitive, these notions remain particularly blurred as for their definition. They have important consequences when sound quantitative assessment of risks and opportunities in face of uncertainty in decision processes are required.

In the course of this lecture, we will study possible mathematical Ansätze to model and quantify these concepts. This will allow us — via robust/dual representation — to better understand what are the key points in the nature of our perception of uncertainty and risk. We, finally, will also address the important problematic of a coherent adaptation of the risk perception as time goes by and which consequence it has in terms of quantification.

Though adopting an abstract “top-down” approach, we will continuously link and discuss the relevance of our construction with many other fields, such as

- Economics: Decision theory, equilibrium…
- Finance: Monetary risk measures, (super)hedging, regulatory rules…
- Psychology: Empirical preference ordering, (ir)rationality…

Mathematically, several fundamental fields will play a role, among others:

- Ordering theory
- Topology
- Functional and convex analysis
- Measure theory
- Stochastic processes

The 4 weekly hours of the course correspond roughly to 3 hours of lecture and 1 hour of exercises. We will cover the following points (the laters, subject to time constraints).

- Preference orders, their numerical representations, Debreu’s Gap-Theorem
- Robust representation of risk measures (Bipolar and Fenchel-Moreau theorem towards a dual representation of quasiconvex lower semicontinuous functionals)
- Application and interpretation for different setups (random variables, probability distributions, von Neumann and Morgenstern representationldots)
- Incomplete preferences, conditional quantification of risk
- Dynamic consistency

Even if we will introduce the necessary mathematical concepts and prove all the results, some knowledge in the following topics is desirable:

- Functional analysis and topology (Hahn-Banach Theorem, some notions of duality, Fenchel-Moreau Theorem is a plus).
- Probability Theory I & II (Financial Mathematics is a plus)

Therefore, the lecture is more adapted to master students.

[1] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: a Hitchhiker’s Guide, Springer Berlin Heidelberg New York, 3 ed., 2006.

[2] P. Fishburn, Nonlinear Preference and Utility Theory, Johns Hopkins University Press, Baltimore, 1988.

[3] H. Follmer and A. Schied, Stochastic Finance. An Introduction in Discrete Time, de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, New York, 2 ed., 2004.