In one-period risk measurement, the risky objects are future net worths modelled by elements of a linear space of real-valued random variables and a risk measure is a mapping from this space of random variables to the real numbers. Typical examples of future net worths are the accounting value of a firm's equity, the surplus of an insurance company or the market value of a portfolio of financial securities. In a dynamic setup one can model the future evolution of net worths with stochastic processes and risks can be calculated at initial and later times. A process of risk measures is called time-consistent if it assigns to a future financial position the same risk irrespective of whether it is calculated directly or in two steps backward in time. The purpose of this project is to understand the structure of time-consistent processes of monetary risk measures, find good examples and study their implementation on a computer. It involves methods from the general theory of stochastic processes, locally convex vector spaces as well as techniques from linear and convex optimization. Monetary risk measures are formally strongly related to pricing in incomplete markets, hedging under constraints, maxmin expected utility functionals, exact games in cooperative game theory as well as to coherent lower previsions in the theory of imprecise probabilities. Since in most of these theories the dynamical aspects are not elaborated yet, research on the dynamics of monetary risk measures will also produce new results in other areas of mathematics and economics.
The proposed work is expected to lead to a new class of risk measures and algorithms for their numerical calculations. The new risk measures will be time-consistent and have good properties for the purposes of pricing, allocation of risk capital and fair premium calculation. This will lead to better methods for the regulation of the banking and insurance industry, the internal risk management of corporations and a realistic assessment of risks the society bears through federal agencies like the Pension Benefit Guaranty Corporation, which guarantees pension benefits promised by insolvent pension plans.
