Refracted Lévy processes

By Andreas E. Kyprianou, Ronnie Loeffen. 2008

68 citations recorded by the source corpus.

Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Levy processes. The latter is a Levy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Levy process is described by the unique strong solution to the stochastic differential equation dU(t) = -delta 1({Ut > b})dt + dX(t), where X = {X-t: t >= 0) is a Levy process with law P and b, delta is an element of R such that the resulting process U may visit the half line (b, infinity) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Levy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.

Subfields: Finance, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6b5cc23bc4e35ce4ccddc876d259de7f4f380a65

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Citations in the corpus

A note on scale functions and the time value of ruin for Levy insurance risk processes (2010, unknown): “…and have proved to be key in understanding optimal barrier strategies (see: Zhou, 2005; Renaud and Zhou, 2007; Kyprianou and Palmowski, 2007; Albrecher, Renaud and Zhou, 2008; Kyprianou and Zhou, 2009; Loeffen, 2009; Kyprianou, Rivero and Song, 2008b; Kyprianou and Loeffen, 2008; Loeffen, 2008a,b).”
ON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY POSITIVE LÉVY PROCESS (2013, background): “…Lévy process X, note that −X is a spectrally negative Lévy process, the existence of unique strong solution to above equation follows directly from Kyprianou and Loeffen (2010), in which the so-called refracted Lévy processes were established for spectrally negative Lévy processes.”
Occupation Times of Refracted Lévy Processes (2012, unknown): “Now using Lemma 20 in [11] we know that Wn(x) →W (x) as n→ ∞, for all x > 0, and also that W ′ n →W ′ as n→ ∞ for almost all x > 0.”
General tax Structures and the Lévy Insurance Risk Model (2009, background): “1) constitute a subclass of controlled Lévy risk processes, the latter being of popular interest in recent literature; see, for example, [5], [13], and [14].”
Occupation times of intervals until first passage times for spectrally negative Lévy processes (2012, methodology): “The main difference between this paper and the papers [9] and [8] is that by using some of the techniques in [7], we find considerably simpler expressions, which further allow us to establish a more general set of identities involving occupation times of spectrally negative Lévy processes.”