CAREER: Quantification of Risk

Princeton University, Princeton NJ

Investigators

Abstract

Quantification of Risk: Abstract The notion of risk plays an important role in various fields such as insurance mathematics, finance, reliability theory, decision theory, economics, optimal control or optimal route planning. In many of these fields, the application of probabilistic and statistical methods has had a long history. A lot of effort has been spent on the development of plausible stochastic models and techniques to maximize expected values, minimize variances or control ruin probabilities. Considerably less work has been devoted to a systematic investigation of the concept of risk, its quantification and the implications for regulation and behavior. In particular, the study of consistent aggregation of risk, attribution of risk, or optimal management of risk in dynamic setups has only started recently. The goal of this project is to provide a mathematical framework for the quantification and management of risk in different contexts. Formally, a risk measure is a functional on a space of random variables or stochastic processes. An important aspect of the project will be the study of the mathematical structure of such functionals with a view towards important properties like monotonicity, convexity or time-consistency. But also the tractability of related optimization problems, practical applications and numerical implementations will be investigated. This will involve methods from probability theory, statistics, convex analysis, locally convex vector spaces and numerical mathematics. The project explores the quantification and management of risks faced by individuals, companies and society in different situations of uncertainty. The main focus of the project is to provide mathematical tools for the assessment, control, mitigation and minimization of risk. Practical problems to be addressed are optimal portfolio allocation; the pricing, hedging and design of financial derivatives; the valuation of complex insurance products such as equity-linked insurance contracts; the determination of capital requirements for banks and insurance companies; coherent attribution of risks; optimal risk transfer between financial institutions; the securitization of risks; integrated risk management; the valuation of energy contracts; the optimal operation of power plants; and the assessment of risk faced by society through financial crises, under-funded pension plans or natural disasters such as hurricanes and earthquakes. The general treatment of risk and the investigation of different concrete problems are expected to motivate new concepts and the transfer of expertise between different fields.