The Theory of Scale Functions for Spectrally Negative Lévy Processes

By Alexey Kuznetsov, Andreas E. Kyprianou, V. Rivero. 2011

297 citations recorded by the source corpus.

The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative Levy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Levy processes, in particular a reasonable understanding of the Levy–Khintchine formula and its relationship to the Levy–Ito decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Levy processes; (Bertoin, Levy Processes (1996); Sato, Levy Processes and Infinitely Divisible Distributions (1999); Kyprianou, Introductory Lectures on Fluctuations of Levy Processes and Their Applications (2006); Doney, Fluctuation Theory for Levy Processes (2007)), Applebaum Levy Processes and Stochastic Calculus (2009).

Subfields: Management Science and Operations Research, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/21428ee8c038c678c8f5ff70a7238bda40a3891a

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Citations in the corpus

Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes (2017, methodology): “By (21) in Kuznetsov et al. (2012), we have Ex[ sup 0≤t≤T1 Xt] = x+ E[ sup 0≤t≤T1 Xt] = x+ µ γ + 1 Φ(γ) , where Φ(γ) = sup{s ≥ 0 : ψ(s) = γ}.”
On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function (2011, background): “[24], Chapters 2, 3, for proofs and references.”
An Insurance Risk Model with Parisian Implementation Delays (2013, unknown): “Further information, examples and numerical techniques related to the computation of scale functions can be found in Kuznetsov et al. (2012).”
Occupation Times of Refracted Lévy Processes (2012, unknown): “19) of [7] that W (θ)′ n (u) W (θ) n (u) = ηn n (ǫ ≥ u) + Φn(θ) (3.”
On magnitude, asymptotics and duration of drawdowns for L\'{e}vy models (2015, background): “ = Z(q)(x) −Z(q)(a) W(q)(x) W(q)(a) , (2.5) where Z(q)(x) = 1 +q Rx 0 W (q)(y)dy. The following lemma gives the behavior of scale functions at 0+ and ∞; see, e.g., Lemmas 3.1, 3.2 of Kuznetsov et al. [18]. Relation (2.6) is from (3.13) of Egamian et al. [14]. Lemma 2.1 For any q≥ 0, W(q)(0+) = ˆ 0, ”
Occupation densities in solving exit problems for Markov additive processes and their reflections (2011, background): “One of the main contribution of this paper is in identification, in Theorem 1, of the scale matrix W which plays the same role as the scale function in the theory of fluctuations of Lévy processes (for a beautiful review on this subject see [1]).”
On pre-exit joint occupation times for spectrally negative Lévy processes (2014, background): “We refer to Kuznetsov et al. (2013) for more detailed discussions and examples of scale functions.”