Fluctuations of Lévy Processes with Applications

Abstract

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Functional limit theorems for processes pieced together from excursions

“…Set S ′′ = (0, ∞), µ(dy) = dy and J n y = c −αn J(c α(κ−β)n y). By (4.11), we see that 14) which shows that (C5) and (C6) are satisfied. We can thus apply Theorem 3.7.…”

“…This is because the functional T 0 : D → [0, ∞] is not continuous; for instance, the sequence of functions w n ∈ D R defined by 14) satisfies that w n converges to…”

Occupation Times of Intervals Until Last Passage Times for Spectrally Negative Lévy Processes

“…The reader is referred to the books Bertoin [3] and Kyprianou [9] for an introduction to the theory of Lévy processes. Here we exclude the case that X is the negative of a subordinator.…”

“…Applying the fluctuation theory and a change of measure, the following results concerning an SNLP can be found from [9] Proposition 1. For x ≥ 0, q ≥ 0, regarding one-sided passage times we have…”

“…For a surplus modelled by a spectrally negative Levy process (SNLP) which we shall denote by X = (X t ) t≥0 , the features at the time of ruin have been throughly studied, see for example Kyprianou [9]. There are also situations, as studied by [4,14,2,1,5], where the moment of ruin may not be the most important character of a risk process.…”