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Lévy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is justified by their application in many areas of classical and modern stochastic models. This textbook forms the basis of a graduate course on the theory and(More)
We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to −∞ a.s. and the positive tail of the Lévy measure, or of the ladder height measure, is subexponential or, more generally,(More)
We obtain a new fluctuation identity for a general Lévy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the under-shoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Klüppelberg, Kyprianou and Maller [Ann. concerning(More)
In this short communication, some of the recent results of Liu (1998) and Biggins and Kyprianou (1997), concerning solutions to a certain functional equation associated with the branching random walk, are strengthened. Their importance is emphasized in the context of travelling wave solutions to a discrete version of the KPP equation and the connection with(More)
The Kesten–Stigum theorem for the one-type Galton–Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton– Watson(More)
Following recent developments in Hubalek and Kyprianou [28] the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative Lévy processes which are completely explicit. This will follow as a consequence of an observation in the aforementioned paper which permits feeding the theory of(More)