Risk, Choquet Capacity and PDE's

Assessing and hedging a future uncertain liability is one of the central questions of finance. In 1999, Artzner, Delbaen, Eber and Heath provided an axiomatic approach which gave the definition of "coherent risk measures". Later Foellmer and Schied, relaxed the positive homogeniety to define the "convex risk measures".

Mathematically, these are maps, with certain properties, which assign a numeric value to each random variable representing the future liability. One may think of them as the capital requirement for carrying the financial positions. Also, they turn out to be nothing but classical utility functions (up to a sign convention) with the additional property called "cash-invariance". In the Markovian situations, these are closely related to nonlinear, scalar, parabolic partial equations. Under the assumptions of the early studies, the resulting equations are semilinear. Recently, this assumptions have been relaxed to allow convex but fully nonlinear equations.

In this talk, I will outline the theory of risk measures, backward stochastic differential equations and their connections to PDEs. This is joint work with Nizar TOUZI (Ecole Polytechnique, Paris) and Jianfeng ZHANG (UCS).