Two-Sided Bounds for Ruin Probabilities when the Adjustment Coefficient does not Exist

Cai, Jun; Garrido, José

doi:10.1080/03461230050131894

Cited by 24 publications

(25 citation statements)

References 25 publications

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“…The following result generalizes a result by Cai and Garrido (1999a), concerning two-sided bounds for the compound geometric tail (and thus for the ruin probability in the renewal risk model) ψ(x).…”

Section: Two-sided Bounds Using a Truncated Lundberg Conditionmentioning

confidence: 59%

“…Then κ x exists uniquely for every positive x and, if κ exists, lim x→∞ κ x = κ (Cai and Garrido, 1999a). The following result generalizes a result by Cai and Garrido (1999a), concerning two-sided bounds for the compound geometric tail (and thus for the ruin probability in the renewal risk model) ψ(x).…”

Section: Two-sided Bounds Using a Truncated Lundberg Conditionmentioning

confidence: 60%

“…(ii) Using again (18), it is immediate to check that we can similarly obtain improvements of the other results in Section 2 in Cai and Garrido (1999a); we have chosen to use Corollary 3 of that paper here because the bounds there are simpler and easier to use.…”

Section: Two-sided Bounds Using a Truncated Lundberg Conditionmentioning

confidence: 93%

See 2 more Smart Citations

Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model

Chadjiconstantinidis¹,

Politis²

2007

Insurance: Mathematics and Economics

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Section: Two-sided Bounds Using a Truncated Lundberg Conditionmentioning

confidence: 59%

Section: Two-sided Bounds Using a Truncated Lundberg Conditionmentioning

confidence: 60%

See 1 more Smart Citation

Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model

Chadjiconstantinidis¹,

Politis²

2007

Insurance: Mathematics and Economics

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“…In particular, in Table 17, the bound U CP is the upper bound for the ruin probability given in Theorem 4.5 of Chadjiconstantinidis and Politis (2005), while the bounds U 1 , U 2 and U 3 are those obtained from (36) for n = 1, 2, 3 and using U (u) = U CP (u) as our initial bound. The last column gives the upper bound from Cai and Garrido (1999). For small values of u, our upper bounds perform better than the initial upper bound U CP (u).…”

Section: Numerical Illustrationsmentioning

confidence: 82%

Tail bounds for the joint distribution of the surplus prior to and at ruin

Psarrakos

Politis

2008

Insurance: Mathematics and Economics

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Keywords: Ruin probability, Time of ruin, Surplus process, Deficit at ruin, Defective renewal equation.We consider the classical risk model where claims Y 1 , Y 2 , . . . arrive in a compound Poisson process with rate λ. The claims are independent identically distributed non-negative random variables and have common distribution function P with finite mean µ. In the case where P has a density, we denote this density by p. We further assume that the claims are independent of the claimarrivals process. Premiums are paid to the insurer continuously at a rate c per unit time. The surplus of the insurer at time t is then U (t) = u + ct − Nt k=1 Y i , where u is the initial surplus and N t is the number of claims until t. We assume throughout that c > λµ, so that ruin is not certain to occur. Moreover, we write c = (1+θ)λµ, where θ is the relative security loading. Let T denote the time of ruin, i.e. the time that the surplus becomes negative for the first time and note that T is a defective random variable. The probability of ruin is then defined by ψ(u) = P (T < ∞|U (0) = u).(1)

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“…If u/β k+1 > 0, then we proceed as follows. According to Cai and Garrido (1999), if q ∈ (0, 1) and 0 < s < ∞ then…”

Section: This Is Equal Tomentioning

confidence: 99%

Modelling heavy-tails with phase-type scale mixture distributions

Xie¹

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In this thesis, we present the new results found in exploring the asymptotic tail probability of the phase-type scale mixture distributions, and apply a subclass of the phase-type scale mixtures, the class of Erlangized scale mixture distributions, to the approximation of the probability of ruin.We consider the class of phase-type scale mixture distributions, which can be written as MellinStieltjes convolution Π G of a phase-type distribution G and a nonnegative but otherwise arbitrary distribution Π. Such a class can also be seen as the class of distributions of the product of two independent random variables: a phase-type random variable Y ∼ G and a nonnegative random variable S ∼ Π. We call S the scaling random variable and its corresponding distribution Π the scaling distribution. We investigate conditions for such a class of distributions to be either light-or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction.In this thesis, particular focus is on phase-type scale mixture distributions where the scaling distribution has discrete support -such a class has been recently used in risk applications to approximate heavy-tailed distributions. We extend an existing scheme for numerically calculating the probability of ruin of a classical Cramér-Lundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a subclass of the phasetype scale mixtures, which is also a dense class of distributions that we denominate Erlangized scale mixtures (ESM). Such a class corresponds to nonnegative and absolutely continuous distributions which can be written as a Mellin-Stieltjes convolution Π G m of a discrete nonnegative distribution Π with an Erlang distribution G m . A distinctive feature of such a class is that it contains heavy-tailed distributions when the scaling distributions have unbounded support, and it provides sharp approximations to heavy-tailed distributions.We suggest a simple methodology for constructing a sequence of distributions having the form Π G m with the purpose of approximating the integrated tail distribution of the claim sizes.Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.ii

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Two-Sided Bounds for Ruin Probabilities when the Adjustment Coefficient does not Exist

Cited by 24 publications

References 25 publications

Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model

Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model

Tail bounds for the joint distribution of the surplus prior to and at ruin

Modelling heavy-tails with phase-type scale mixture distributions

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