The monopolist's problem of deciding what types of products to manufacture and how much to charge for each of them, knowing only statistical information about the preferences of an anonymous field of potential buyers, is one of the basic problems analyzed in economic theory. The solution to this problem when the space of products and of buyers can each be parameterized by a single variable (say quality X, and income Y) garnered Mirrlees (1971) and Spence (1974) their Nobel prizes in 1996 and 2001. The multidimensional version of this question is a largely open problem in the calculus of variations (see Basov's book "Multidimensional Screening".) I will describe recent work with A Figalli and Y-H Kim, identifying structural conditions on the value b(X,Y) of product X to buyer Y which reduce this problem to a convex program in a Banach space--- leading to uniqueness and stability results for its solution, confirming robustness of certain economic phenomena observed by Armstrong (1996) such as the desirability for the monopolist to raise prices enough to drive a positive fraction of buyers out of the market, and yielding conjectures about the robustness of other phenomena observed Rochet and Chone (1998), such as the clumping together of products marketed into subsets of various dimension. The passage to several dimensions relies on ideas from differential geometry / general relativity, optimal transportation, and nonlinear PDE.

• Thematic program: Financial Engineering for Energy Asset Management and hedging in commodity markets (ENERGY-10-Prolongation) (2010 Prolongation into 2011)

In the second part of this mini-course we will introduce jump processes (or inhomogeneous Levy processes) for modelling the dynamics of energy prices. We analyse multi-factor Ornstein-Uhlenbeck processes with stochatsic volatility as a general class of spot price models, and link these to forward prices. Our models will be motivated by stylized facts of energy prices, like mean-reversion, seasonality and spikes. Finally, we study pricing of options on forwards, based on Fourier methods analysed in detail in the first part of the mini-course by Professor Eberlein.

• Event: Mini-course on "Fourier methods in mathematical finance with applications to Energy and Commodity markets" (2012)

The aim of this mini-course is to provide a systematic analysis of valuation formulas for derivatives in finance which are based on Fourier transforms. In the first part we concentrate on the case where the underlying security is modeled by an exponential semimartingale in general. This covers e.g. stock prices, indices and FX rates. In particular Lévy processes as drivers are studied in detail. A great variety of payoff functions and specific processes can be considered within this framework. Formulas for derivatives which depend on multidimensional underlyers are considered as well. The Fourier based approach allows also to compute Greeks.
In the second part of this mini-course we will introduce jump processes (or inhomogeneous Levy processes) for modelling the dynamics of energy prices. We analyse multi-factor Ornstein-Uhlenbeck processes with stochatsic volatility as a general class of spot price models, and link these to forward prices. Our models will be motivated by stylized facts of energy prices, like mean-reversion, seasonality and spikes. Finally, we study pricing of options on forwards, based on Fourier methods analysed in detail in the first part of the mini-course by Professor Eberlein.

• Event: Mini-course on "Fourier methods in mathematical finance with applications to Energy and Commodity markets" (2012)