Extremal behavior of stochastic integrals driven by regularly varying Lévy processes

We study the extremal behavior of a stochastic integral driven by a multivariate Lévy process that is regularly varying with index α > 0. For predictable integrands with a finite (α + δ)-moment, for some δ > 0, we show that the extremal behavior of the stochastic integral is due to one big jump of the driving Lévy process and we determine its limit measure associated with regular variation on the space of c`adì ag functions. 1. Introduction. Stochastic integrals driven by Lévy processes constitute a broad and popular class of semimartingales used as the driving noise in a wide variety of probabilistic models, for instance, the evolution of assets prices in mathematical finance. The extremal behavior of these processes is of importance when computing failure probabilities in various systems, for example, the probability that a functional of the sample path of the process exceeds some high threshold. In the presence of heavy tails of the underlying noise process such failures are often most likely due to one or a few unlikely events, such as large discontinuities (jumps) of the driving noise process. In the presence of Pareto-like tails of the underlying distributions regular variation on the space of c`adì ag functions provides a useful framework to describe the extremal behavior of stochastic processes and approximate failure probabilities. In this paper we study the extremal behavior of stochastic in-tegrals with respect to regularly varying Lévy processes. A first step toward studying the extremes of these processes was communicated to the authors by D. Applebaum [2]. The notion of regular variation is fundamental in various fields of applied probability. It serves as domain of attraction condition for partial sums of