Refracted Lévy processes

Kyprianou, Andreas E.; Loeffen, Ronnie

doi:10.1214/08-aihp307

Cited by 87 publications

(252 citation statements)

References 29 publications

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“…We obtained relatively simple expressions for the special but important case that I(x) = αx. In this case the results are in the same flavor as for spectrally negative refracted Lévy processes, where the premium income rate when the reserve is above γ is αc, as was studied by Kyprianou and Loeffen (2010).…”

Section: Discussionmentioning

confidence: 78%

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A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function

2017

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We consider a Cramér-Lundberg insurance risk process with the added feature of reinsurance. If an arriving claim finds the reserve below a certain threshold γ, or if it would bring the reserve below that level, then a reinsurer pays part of the claim. Using fluctuation theory and the theory of scale functions of spectrally negative Lévy processes, we derive expressions for the Laplace transform of the time to ruin and of the joint distribution of the deficit at ruin and the surplus before ruin. We specify these results in much more detail for the threshold set-up with proportional reinsurance.

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Section: Discussionmentioning

confidence: 78%

“…We apply similar methods as in Kyprianou and Loeffen (2010) to obtain similar quantities of interest. Assume that the company has a reinsurance contract.…”

Section: Introductionmentioning

confidence: 99%

A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function

2017

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“…Lévy processes without positive jumps and with state-dependent exponents are also important in finance, where they are sometimes called refracted Lévy processes (see [23]). In this area, contrary to the queueing and dam applications, the processes are typically not reflected at 0 and we are interested in first-exit problems instead of steady-state distributions.…”

Section: Introductionmentioning

confidence: 99%

“…This may be relevant for financial models; see, e.g. [1, Chapter XIV] or [2], [18], [19], [23], [26], [30], and the references therein. The solution procedure then has to be slightly adapted.…”

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confidence: 99%

Lévy processes with adaptable exponent

Bekker

Boxma²,

Resing³

2009

Adv. Appl. Probab.

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In this paper we consider Lévy processes without negative jumps, reflected at the origin. Feedback information about the level of the Lévy process ('workload level') may lead to adaptation of the Lévy exponent. Examples of such models are queueing models in which the service speed or customer arrival rate changes depending on the workload level, and dam models in which the release rate depends on the buffer content. We first consider a class of models where information about the workload level is continuously available. In particular, we consider dam processes with a two-step release rule and M/G/1 queues in which the arrival rate, service speed, and/or jump size distribution may be adapted depending on whether the workload is above or below some level K. Secondly, we consider a class of models in which the workload can only be observed at Poisson instants. At these Poisson instants, the Lévy exponent may be adapted based on the amount of work present. For both classes of models, we determine the steady-state workload distribution.

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“…Other advantages of scale functions can be seen in recent literature that looks at dividend payments (for example in the form of reflection or refraction strategies) 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). In the context of last ruin times Chiu and Yin (2005) and Baurdoux (2008) also make extensive use of the theory of scale functions.…”

Section: Introductionmentioning

confidence: 99%

A Note on Scale Functions and the Time Value of Ruin for Lévy Insurance Risk Processes

Biffis

Kyprianou

2009

SSRN Journal

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Abstract. We examine discounted penalties at ruin for surplus dynamics driven by a general spectrally negative Lévy process; the natural class of stochastic processes which contains many examples of risk processes which have already been considered in the existing literature. Following from the important contributions of Zhou (2005) we provide an explicit characterization of a generalized version of the Gerber-Shiu function in terms of scale functions, streamlining and extending results available in the literature.

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Refracted Lévy processes

Cited by 87 publications

References 29 publications

A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function

A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function

Lévy processes with adaptable exponent

A Note on Scale Functions and the Time Value of Ruin for Lévy Insurance Risk Processes

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