The surplus prior to ruin and the deficit at ruin for a correlated risk process

Badescu, Andrei L.; Breuer, Lothar; Drekic, Steve; Latouche, Guy; Stanford, David A.

doi:10.1080/03461230510009835

Cited by 25 publications

(36 citation statements)

References 10 publications

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“…Between the three parts we need to take possible (and necessary) phase changes into account. This reasoning will be similar to [8]. A difference to [13] is that we measure the timesGτ (u) andτ (u) −Gτ (u) in terms of their Laplace transforms.…”

Section: Remarkmentioning

confidence: 99%

“…The joint density function of the undershoot and overshoot has been derived in [8] for the fluid flow case from an insurance perspective. We setX t := u − U t , where u is the initial capital and U t is the risk reserve process as given in [8, p. 434].…”

Section: Examplementioning

confidence: 99%

“…for γ ≥ 0, where φ(β) and φ * (β) for β > 0 are given in the last line of (6) and in (8), respectively.…”

Section: Then the Pairs (G E (Q) B E (Q) ) And (Gmentioning

confidence: 99%

“…Mostly, it has been investigated within the framework of the so-called Gerber-Shiu (discounted penalty) function, which describes the joint distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin. Some examples of stochastic processes for which results are available are the compound Poisson model [16], its perturbed [12], [15], [22] and Markov-modulated [4], [25] versions, the Lévy risk process [14], the fluid flow model [1], [8], the Sparre Andersen model with Erlang interclaim times [17] as well as its perturbed version [20]. This list is by no means exhaustive, but one can already see that all these models are special instances of Markov additive processes (MAPs).…”

Section: Introductionmentioning

confidence: 99%

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A quintuple law for Markov additive processes with phase-type jumps

Breuer¹

2010

J. Appl. Probab.

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We consider a Markov additive process (MAP) with phase-type jumps, starting at 0. Given a positive level u, we determine the joint distribution of the undershoot and overshoot of the first jump over the level u, the maximal level before this jump, the time of attaining this maximum, and the time between the maximum and the jump. The analysis is based on first passage times and time reversion of MAPs. A marginal of the derived distribution is the Gerber-Shiu function, which is of interest to insurance risk. Several examples serve to compare the present result with the literature.

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Section: Remarkmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

“…for γ ≥ 0, where φ(β) and φ * (β) for β > 0 are given in the last line of (6) and in (8), respectively.…”

Section: Then the Pairs (G E (Q) B E (Q) ) And (Gmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A quintuple law for Markov additive processes with phase-type jumps

Breuer¹

2010

J. Appl. Probab.

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“…In Bean et al [12], Y (t) represents the amount of wear of a power generator and it may jump from B (indicating that the generator is unusable) to 0 (indicating that it has been replaced by a new equipment). As discussed in Badescu et al [7] and Stanford et al [47], risk processes may be analysed as fluid queues with jumps.…”

Section: Extensionsmentioning

confidence: 99%

Analysis of Fluid Flow Models

Dattatreya¹

2008

Chapman &Amp; Hall/CRC Computer &Amp; Information Science Series

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Markov-modulated fluids have a long history. They form a simple class of Markov additive processes, and were initially developed in the 1950s as models for dams and reservoirs, before gaining much popularity in the 1980s as models for buffers in telecommunication systems, when they became known as fluid queues. More recent applications are in risk theory and in environmental studies.In telecommunication systems modelling, the attention focuses on determining the stationary distribution of the buffer content. Early ODE resolution techniques have progressively given way to approaches grounded in the analysis of the physical evolution of the system, and one only needs now to solve a Riccati equation in order to obtain several quantities of interest. To the early algorithms proposed in the Applied Probability literature, numerical analysts have added new algorithms, improved in terms of convergence speed, numerical accuracy, and domain of applicability.We give here a high-level presentation of the matrix-analytic approach to the analysis of fluid queues, briefly address computational issues, and conclude by indicating how this has been extended to more general processes.

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New Research Directions in Modern Actuarial Sciences

Bulinskaya

2017

Springer Proceedings in Mathematics &Amp; Statistics

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The surplus prior to ruin and the deficit at ruin for a correlated risk process

Cited by 25 publications

References 10 publications

A quintuple law for Markov additive processes with phase-type jumps

A quintuple law for Markov additive processes with phase-type jumps

Analysis of Fluid Flow Models

New Research Directions in Modern Actuarial Sciences

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