The Cramér condition is necessary and sufficient for asymptotically exponential decrease of ruin probability

1999 ◽  
Vol 93 (4) ◽  
pp. 609-611 ◽  
Author(s):  
O. P. Vinogradov
Keyword(s):  
Ruin Probability ◽  
Cramer Condition ◽  
Exponential Decrease ◽  
Cramér Condition ◽  
Necessary And Sufficient

scholarly journals The Pólya-Aeppli process and ruin problems

2004 ◽  
Vol 2004 (3) ◽  
pp. 221-234 ◽  
Author(s):  
Leda D. Minkova
Keyword(s):  
Poisson Process ◽  
Ruin Probability ◽  
Counting Process ◽  
Risk Model ◽  
Classical Risk Model ◽  
Homogeneous Poisson Process ◽  
Cramer Condition ◽  
Cramér Condition

The Pólya-Aeppli process as a generalization of the homogeneous Poisson process is defined. We consider the risk model in which the counting process is the Pólya-Aeppli process. It is called a Pólya-Aeppli risk model. The problem of finding the ruin probability and the Cramér-Lundberg approximation is studied. The Cramér condition and the Lundberg exponent are defined. Finally, the comparison between the Pélya-Aeppli risk model and the corresponding classical risk model is given.

scholarly journals Approximations of Ruin Probability by Di-Atomic or Di-Exponential Claims

1992 ◽  
Vol 22 (2) ◽  
pp. 235-246 ◽  
Author(s):  
Joshua Babier ◽  
Beda Chan
Keyword(s):  
Ruin Probability ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ruin Probabilities ◽  
Claim Size ◽  
Group Life ◽  
Large Spread ◽  
Claim Size Distribution ◽  
Necessary And Sufficient ◽  
Insurance Data

AbstractThe sensitivity of the ruin probability depending on the claim size distribution has been the topic of several discussion papers in recent ASTIN Bulletins. This discussion was initiated by a question raised by Schmitter at the ASTIN Colloquium 1990 and attempts to make further contributions to this problem. We find the necessary and sufficient conditions for fitting three given moments by diatomic and diexponential distributions. We consider three examples drawn from fire (large spread), individual life (medium spread) and group life (small spread) insurance data, fit them with diatomics and diexponentials whenever the necessary and sufficient conditions are met, and compute the ruin probabilities using well known formulas for discrete and for combination of exponentials claim amounts. We then compare our approximations with the exact values that appeared in the literature. Finally we propose using diatomic and diexponential claim distributions as tools to study the Schmitter problem.

A necessary and sufficient condition for the subexponentiality of the product convolution

2018 ◽  
Vol 50 (01) ◽  
pp. 57-73 ◽  
Author(s):  
Hui Xu ◽  
Fengyang Cheng ◽  
Yuebao Wang ◽  
Dongya Cheng
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Sufficient Conditions ◽  
Related Result ◽  
Sufficient Condition ◽  
Insurance Risk ◽  
Necessary And Sufficient Condition ◽  
Long Tail ◽  
Stochastic Returns ◽  
Necessary And Sufficient

Abstract Let X and Y be two independent and nonnegative random variables with corresponding distributions F and G. Denote by H the distribution of the product XY, called the product convolution of F and G. Cline and Samorodnitsky (1994) proposed sufficient conditions for H to be subexponential, given the subexponentiality of F. Relying on a related result of Tang (2008) on the long-tail of the product convolution, we obtain a necessary and sufficient condition for the subexponentiality of H, given that of F. We also study the reverse problem and obtain sufficient conditions for the subexponentiality of F, given that of H. Finally, we apply the obtained results to the asymptotic study of the ruin probability in a discrete-time insurance risk model with stochastic returns.

scholarly journals Stability of the exit time for Lévy processes

2011 ◽  
Vol 43 (03) ◽  
pp. 712-734
Author(s):  
Philip S. Griffin ◽  
Ross A. Maller
Keyword(s):  
Lévy Process ◽  
Exit Time ◽  
Passage Time ◽  
Risk Process ◽  
Levy Processes ◽  
Levy Process ◽  
Conditional Stability ◽  
Insurance Risk ◽  
Cramer Condition ◽  
Cramér Condition

This paper is concerned with the behaviour of a Lévy process when it crosses over a positive level, u, starting from 0, both as u becomes large and as u becomes small. Our main focus is on the time, τ u , it takes the process to transit above the level, and in particular, on the stability of this passage time; thus, essentially, whether or not τ u behaves linearly as u ↓ 0 or u → ∞. We also consider the conditional stability of τ u when the process drifts to -∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process, which we analyse under a Cramér condition.

scholarly journals Large deviation results for a U-statistical sum with product kernel

2001 ◽  
Vol 63 (1) ◽  
pp. 151-165 ◽  
Author(s):  
Y. V. Borovskikh ◽  
N. C. weber
Keyword(s):  
Large Deviation ◽  
Product Kernel ◽  
Cramer Condition ◽  
Cramér Condition

Large deviation theorems are proved for non-degenerate U-statistical sums of degree m with kernel h (x1, …, xm) = x1 … xm under the Cramér condition and under the Linnik condition. The method of proof uses truncation and the contraction technique.

scholarly journals Stability of the exit time for Lévy processes

2011 ◽  
Vol 43 (3) ◽  
pp. 712-734 ◽  
Author(s):  
Philip S. Griffin ◽  
Ross A. Maller
Keyword(s):  
Lévy Process ◽  
Exit Time ◽  
Passage Time ◽  
Risk Process ◽  
Levy Processes ◽  
Levy Process ◽  
Conditional Stability ◽  
Insurance Risk ◽  
Cramer Condition ◽  
Cramér Condition

This paper is concerned with the behaviour of a Lévy process when it crosses over a positive level, u, starting from 0, both as u becomes large and as u becomes small. Our main focus is on the time, τu, it takes the process to transit above the level, and in particular, on the stability of this passage time; thus, essentially, whether or not τu behaves linearly as u ↓ 0 or u → ∞. We also consider the conditional stability of τu when the process drifts to -∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process, which we analyse under a Cramér condition.

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