The surpluses immediately before and at ruin, and the amount of the claim causing ruin

Dufresne, François; Gerber, Hans U.

doi:10.1016/0167-6687(88)90076-5

Cited by 81 publications

(43 citation statements)

References 2 publications

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“…In view of the results of Dufresne and Gerber [11], it is tempting to believe that τ − − T N − has an exponential distribution. However, it turns out that this 'undershoot' has a completely different distribution.…”

Section: A Single Compound Poisson Inputmentioning

confidence: 99%

Quasi-Product Forms for Lévy-Driven Fluid Networks

2007

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We study stochastic tree fluid networks driven by a multidimensional Lévy process. We are interested in (the joint distribution of) the steady-state content in each of the buffers, the busy periods, and the idle periods. To investigate these fluid networks, we relate the above three quantities to fluctuations of the input Lévy process by solving a multidimensional Skorokhod reflection problem. This leads to the analysis of the distribution of the componentwise maximums, the corresponding epochs at which they are attained, and the beginning of the first last-passage excursion. Using the notion of splitting times, we are able to find their Laplace transforms. It turns out that, if the components of the Lévy process are 'ordered', the Laplace transform has a so-called quasi-product form.The theory is illustrated by working out special cases, such as tandem networks and priority queues.

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Section: A Single Compound Poisson Inputmentioning

confidence: 99%

Quasi-Product Forms for Lévy-Driven Fluid Networks

2007

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“…For such processes, many risk measures have been considered (see, for example, Gerber (1988), Dufresne and Gerber (1988), and Picard (1994)): the time to ruin T u = inf{t > 0 : u + X t < 0}; the severity of ruin u + X T u ; the pair (T u , u + X T u ); the time in the red (that is, below 0) T u − T u from the time of first ruin to the time of first recovery, where T u = inf{t > T u : u + X t = 0}; the maximal ruin severity inf t>0 (u + X t ); the aggregate severity of ruin until recovery J (u) = T u T u |u + X t | dt; etc. Dos Reis (1993) studied the total time in the red τ (u) = ∞ 0 1 {u+X t <0} dt, using results of Gerber (1988).…”

Section: Introductionmentioning

confidence: 99%

Differentiation of some functionals of risk processes, and optimal reserve allocation

Loisel¹

2005

J. Appl. Probab.

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For general risk processes, we introduce and study the expected time-integrated negative part of the process on a fixed time interval. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below 0, studied by Dos Reis, and the probability of ruin. We carry out differentiation of other functionals of one-dimensional and multidimensional risk processes with respect to the initial reserve level. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve that minimizes one of these risk measures, illustrate the variety of fields of application and the benefits deriving from an efficient and effective use of such tools.

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“…The probability of ultimate ruin with initial reserves u, parameter λ and severity of ruin less than y and surplus less than x + u is defined, We will now use the results obtained by Dufresne and Gerber (1988) and a similar renewal argument as in Frey and Schmidt (1996) and Gerber et al (1987) in order to express the former probability with the following defective renewal equation: for x ∈ (0, ∞) and y ∈ (0, ∞).…”

Section: Introductionmentioning

confidence: 99%

Calculating multivariate ruin probabilities via Gaver–Stehfest inversion technique

Usabel

1999

Insurance: Mathematics and Economics

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Multivariate characteristics of risk processes are of high interest to academic actuaries. In such models, the probability of ruin is obtained not only by considering initial reserves u but also the severity of ruin y and the surplus before ruin x. This ruin probability can be expressed using an integral equation that can be efficiently solved using the Gaver-Stehfest method of inverting Laplace transforms. This approach can be considered to be an alternative to recursive methods previously used in actuarial literature.JEL classification: G22; IM13; IM20

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The surpluses immediately before and at ruin, and the amount of the claim causing ruin

Cited by 81 publications

References 2 publications

Quasi-Product Forms for Lévy-Driven Fluid Networks

Quasi-Product Forms for Lévy-Driven Fluid Networks

Differentiation of some functionals of risk processes, and optimal reserve allocation

Calculating multivariate ruin probabilities via Gaver–Stehfest inversion technique

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