During the last decade, problems in the world of finance have been the main driving force for developing sophisticated mathematical methods which may be used for identifying and measuring risk. The focus is still on quantifying market and credit risk, but general operational risks will become more important in the future. In this book the reader will find approaches from economic theory, allocation problems, credit scoring, volatility structures, general market risk, country risk and extreme value theory. The contributions of this book reflect the views of leading practitioners and academics in the field of risk management. Most of the models considered for the evolution of asset values are of a complex and stochastic nature, including stochastic volatility models in continuous time as well as their counterparts in discrete time, the family of GARCH-like time series. The contents reflect the fact that a major part of recent research has been motivated by applications in finance, but most of the mathematical approaches may be used for risk analysis in engineering and science in a rather straightforward manner. As known from insurance mathematics for some time, extreme damages from natural disaster follow similar stochastic laws as extreme losses from certain investments. The articles discuss critical concepts such as value-at-risk, volatility and other risk masures in nonstandard situations. Stochastic processes beyond geometric Brownian motion allow for a more realistic reflection of stylized facts like leptokurtosis or skewness of return distrubutions which often are observed in real data. Procedures for detecting change points in time series allow for dealing with the risk of a sudden structural change of the market. Models for extremal events in financial time series or stochastic processes in continuous time are of prime importance for risk management as, in practice, these rare events frequently dominate the whole profit/loss-process.

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