5 ECM Minisymposium Mathematical Finance 
Mathematical finance has been the fastest growing branch of applied mathematics in the past decades. In addition to the very significant impact that this field has on financial markets, there are strong impulses towards mathematics: more detailed studies of stochastic processes, new insights in optimization and duality theory, computational challenges going well beyond the capabilities of traditional methods. Generous support from the ESF program Advanced Mathematical Methods in Finance (AMaMeF) has made it possible to invite four of the world's most prominent researchers in this area to present their work at 5ECM.
Organization: Hans Schumacher (Tilburg University) and Peter Spreij (Universiteit van Amsterdam) 
Date:  July 15, 2008 
Time:  13:25  16:55 
Location:  RAI Congress Center (Travel directions), Room CD 
13:2514:10  Hans Föllmer  Probabilistic Quantification of Financial Uncertainty 
14:1014:55  Philip Protter  Modelling Financial Bubbles 
14:5515:25  
15:2516:10  Walter Schachermayer  The Fundamental Theorem of Asset Pricing for Continuous Processes under Small Transaction Costs 
16:1016:55  Thaleia Zariphopoulou  SPDE and portfolio choice 

Hans Föllmer: Probabilistic Quantification of Financial Uncertainty
We discuss some recent advances in the probabilistic analysis of financial risk under model uncertainty, including risk measures and their dynamics, connections to the microeconomic theory of preferences, and problems of robust portfolio choice. 
Philip Protter: Modelling Financial Bubbles
The mathematical modeling of financial bubbles involves subtle distinctions between types of martingales and strict local martingales. We will discuss how one can obtain the birth of a bubble through the market's shifting choices of risk neutral measures. The presence of bubbles leads to some conventional wisdom being false, such as "no early exercise" with American call options. 
Walter Schachermayer:
The Fundamental Theorem of Asset Pricing for Continuous Processes under Small Transaction Costs
A version of the fundamental theorem of asset pricing is proved for continuous asset prices with small proportional transaction costs. Equivalence is established between: (a) the absence of arbitrage with general strategies for arbitrarily small transaction costs ε >0, (b) the absence of free lunches with bounded risk for arbitrarily small transaction costs ε >0, and (c) the existence of εconsistent price systems  the analogue of martingale measures under transaction costs  for arbitrarily small ε >0. The proof proceeds through an explicit construction, as opposed to the usual separation arguments. The paper concludes comparing numérairefree and numérairebased notions of admissibility, and the corresponding martingale and local martingale properties for consistent price systems. 
Thaleia Zariphopoulou:
SPDE and portfolio choice A new approach to risk preference specification (forward performance processes) and optimal portfolio choice will be presented. The class of SPDE satisfied by the forward processes will be discussed as well as the properties of their solutions. The problem of how risk preferences can be inferred from a desired bespoke investment plan will be also discussed. 