Lévy Matters I

Over the past 10-15 years, we have seen a revival of general Levy ¿ processes theory as well as a burst of new applications. In the past, Brownian motion or the Poisson process have been considered as appropriate models for most applications. Nowadays, the need for more realistic modelling of irregular behaviour of phen- ena in nature and society like jumps, bursts, and extremeshas led to a renaissance of the theory of general Levy ¿ processes. Theoretical and applied researchers in elds asdiverseas quantumtheory,statistical physics,meteorology,seismology,statistics, insurance, nance, and telecommunication have realised the enormous exibility of Lev ¿ y models in modelling jumps, tails, dependence and sample path behaviour. L¿ evy processes or Levy ¿ driven processes feature slow or rapid structural breaks, extremal behaviour, clustering, and clumping of points. Toolsandtechniquesfromrelatedbut disctinct mathematical elds, such as point processes, stochastic integration,probability theory in abstract spaces, and differ- tial geometry, have contributed to a better understanding of Le ¿vy jump processes. As in many other elds, the enormous power of modern computers has also changed the view of Levy ¿ processes. Simulation methods for paths of Levy ¿ p- cesses and realisations of their functionals have been developed. Monte Carlo simulation makes it possible to determine the distribution of functionals of sample paths of Levy ¿ processes to a high level of accuracy.