Dynamic Monetary Risk Measures

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In one-period risk measurement, the risky objects are futurenet worths modelled by elements of a linear space of real-valued random variablesand 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 afirm's equity, the surplus of an insurance company or the marketvalue of a portfolio of financial securities.In a dynamic setup one can model the future evolution of networths with stochastic processes and risks can be calculated atinitial and later times. A process of risk measures is called time-consistent ifit assigns to a future financial position the same risk irrespective of whether it iscalculated directly or in two steps backward in time. The purpose of this projectis to understand the structure of time-consistent processes of monetary risk measures,find good examples and study their implementation on a computer. It involves methodsfrom the general theory of stochastic processes, locally convex vector spaces aswell as techniques from linear and convex optimization.Monetary risk measures are formally strongly relatedto pricing in incomplete markets, hedging under constraints,maxmin expected utility functionals,exact games in cooperative game theory as well as tocoherent lower previsions in the theory of imprecise probabilities.Since in most of these theories the dynamical aspects are not elaboratedyet, research on the dynamics of monetary risk measures willalso produce new results in other areas of mathematics and economics.The proposed work is expected to lead to a new class of riskmeasures and algorithms for their numerical calculations. The newrisk measures will be time-consistent and have good properties for the purposesof pricing, allocation of risk capital and fair premium calculation. This will leadto better methods for the regulation of the banking and insuranceindustry, the internal risk management of corporations and arealistic assessment of risks the society bears through federalagencies like the Pension Benefit Guaranty Corporation, whichguarantees pension benefits promised by insolvent pension plans.