Estimates for the probability of ruin with special emphasis on the possibility of large claims

Embrechts, Paul; Veraverbeke, Noël

doi:10.1016/0167-6687(82)90021-x

Cited by 333 publications

(205 citation statements)

References 9 publications

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“…From this, and the inequality e sz ≤ 1 + sz + s 2 K 2 e sK , valid for any s ≥ 0 and for any z such that |z| ≤ K, it follows that the left side of (21) is bounded above by Eϕ n + sK 2 e sK , which, by (16), is less than or equal to zero for any s > 0 such that sK 2 e sK ≤ −α/4.…”

Section: A Uniform Upper Bound For Discrete-time Processes With Indepmentioning

confidence: 83%

Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments

2007

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We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Lévy process with heavy-tailed Lévy measure. A central point of the paper is that we make full use of the so-called "principle of a single big jump" in order to obtain both upper and lower bounds. Thus, the proofs are entirely probabilistic. The paper is motivated by queueing and Lévy stochastic networks.

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Section: A Uniform Upper Bound For Discrete-time Processes With Indepmentioning

confidence: 83%

Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments

2007

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“…Recent studies of this class can be found in Pakes (2004), Tang (2006), Foss and Korshunov (2007), and Watanabe (2008), among many others. This class is often used to model claim-size distributions; see, for example, Embrechts and Veraverbeke (1982), Klüppelberg (1989a), and Tang and Tsitsiashvili (2004). It is well known that the constant c in relation (2.5) is equal to the moment generating function of F at γ, defined to be…”

Section: On the Claim-size Distributionmentioning

confidence: 99%

Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims

Jiang

Tang

2008

Insurance: Mathematics and Economics

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“…It is well known (see, e.g., Asmussen [1], Embrechts, Klüppelberg and Mikosch [12], Embrechts and Veraverbeke [13] and references therein) that if F I is subexponential, then…”

Section: Supremum Of a Random Walkmentioning

confidence: 99%

On lower limits and equivalences for distribution tails of randomly stopped sums

Denisov¹,

Foss²,

Korshunov³

2008

Bernoulli

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For a distribution F * τ of a random sum Sτ = ξ1 + • • • + ξτ of i.i.d. random variables with a common distribution F on the half-line [0, ∞), we study the limits of the ratios of tails F * τ (x)/F (x) as x → ∞ (here, τ is a counting random variable which does not depend on {ξn} n≥1 ). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.

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Estimates for the probability of ruin with special emphasis on the possibility of large claims

Cited by 333 publications

References 9 publications

Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments

Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments

Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims

On lower limits and equivalences for distribution tails of randomly stopped sums

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