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Queues and Lévy Fluctuation Theory
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The class of Lévy processes consists of all stochastic processes with stationary and independent increments; here ‘stationarity’ means that increments corresponding to a fixed time interval are identically distributed, whereas ‘independence’ refers to the property that increments corresponding to non-overlapping time intervals behave statistically independently. As such, Lévy processes cover several well-studied processes (e.g. Brownian motions and Poisson processes), but also, as this book will show, a wide variety of other processes, with their own specific properties in terms of their path structure. The process’ increments being stationary and independent, Lévy processes can be seen as the genuine continuous-time counterpart of the random walk Sn WD PniD1 Yi, with independent and identically distributed Yi.
Lévy processes owe their popularity to their mathematically attractive properties as well as their wide applicability: they play an increasingly important role in a broad spectrum of application domains, ranging from finance to biology. Lévy processes were named after the French mathematician Paul Lévy (1886–1971), who played a pioneering role in the systematic analysis of processes with stationary and independent increments. A brief account of the history of Lévy processes (initially simply known as ‘processes with stationary and independent increments’) and its application fields is given in e.g. Applebaum [12].
Application areas—In mathematical finance, Lévy processes are being used intensively to analyze various phenomena. They are for instance suitable when studying credit risk, or for option pricing purposes (see e.g. Cont and Tankov [63]), but play a pivotal role in insurance mathematics as well (see e.g. Asmussen and Albrecher [21]). An attractive feature of Lévy processes, particularly with applications in finance in mind, is that this class is rich in terms of possible path structures: it is perhaps the simplest class of processes that allows sample paths to have continuous parts interspersed with jumps at random epochs
Krzysztof De ̨bicki • Michel Mandjes - Personal Name
1st Edtion
978-3-319-20693-6
NONE
Queues and Lévy Fluctuation Theory
Information Technology
English
Springer Cham Heidelberg
2015
USA
1-256
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