scholarly journals Ruin Probability Approximations in Sparre Andersen Models with Completely Monotone Claims

Risks ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 104 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Eleni Vatamidou
Keyword(s):  
Laplace Transform ◽  
Error Bounds ◽  
Efficient Algorithm ◽  
Ruin Probability ◽  
Risk Process ◽  
Claim Size ◽  
Size Distributions ◽  
Completely Monotone ◽  
Sparre Andersen Risk Process

We consider the Sparre Andersen risk process with interclaim times that belong to the class of distributions with rational Laplace transform. We construct error bounds for the ruin probability based on the Pollaczek–Khintchine formula, and develop an efficient algorithm to approximate the ruin probability for completely monotone claim size distributions. Our algorithm improves earlier results and can be tailored towards achieving a predetermined accuracy of the approximation.

scholarly journals On the expected time to ruin and the expected dividends when dividends are paid while the surplus is above a constant barrier

2005 ◽  
Vol 42 (3) ◽  
pp. 595-607 ◽  
Author(s):  
Esther Frostig
Keyword(s):  
Ruin Probability ◽  
Risk Process ◽  
Claim Size ◽  
Compound Poisson ◽  
Expected Time ◽  
Premium Rate ◽  
Phase Type

We study the expected time to ruin in a risk process in which dividends are paid when the surplus is above the barrier. We consider the case in which the dividend rate is smaller than the premium rate. We obtain results for the classical compound Poisson risk process with phase-type claim size. When the ruin probability is 1, we derive the expected time to ruin and the expected dividends paid. When the ruin probability is less than 1, these quantities are derived conditioning on the event that ruin occurs.

scholarly journals On the expected time to ruin and the expected dividends when dividends are paid while the surplus is above a constant barrier

2005 ◽  
Vol 42 (03) ◽  
pp. 595-607 ◽  
Author(s):  
Esther Frostig
Keyword(s):  
Ruin Probability ◽  
Risk Process ◽  
Claim Size ◽  
Compound Poisson ◽  
Expected Time ◽  
Premium Rate ◽  
Phase Type

We study the expected time to ruin in a risk process in which dividends are paid when the surplus is above the barrier. We consider the case in which the dividend rate is smaller than the premium rate. We obtain results for the classical compound Poisson risk process with phase-type claim size. When the ruin probability is 1, we derive the expected time to ruin and the expected dividends paid. When the ruin probability is less than 1, these quantities are derived conditioning on the event that ruin occurs.

Expansions for Markov-modulated systems and approximations of ruin probability

1996 ◽  
Vol 33 (01) ◽  
pp. 57-70
Author(s):  
Bartłomiej Błaszczyszyn ◽  
Tomasz Rolski
Keyword(s):  
Ruin Probability ◽  
Marked Point ◽  
Factorial Moment ◽  
Risk Process ◽  
Rate Function ◽  
Marked Point Process ◽  
Probability Of Ruin ◽  
Actual Risk ◽  
Moment Measures ◽  
Markov Modulated

Let N be a stationary Markov-modulated marked point process on ℝ with intensity β ∗ and consider a real-valued functional ψ(N). In this paper we study expansions of the form Eψ(N) = a 0 + β ∗ a 1 + ·· ·+ (β∗ ) nan + o((β ∗) n ) for β ∗→ 0. Formulas for the coefficients ai are derived in terms of factorial moment measures of N. We compute a 1 and a 2 for the probability of ruin φ u with initial capital u for the risk process in the Markov-modulated environment; a 0 = 0. Moreover, we give a sufficient condition for ϕu to be an analytic function of β ∗. We allow the premium rate function p(x) to depend on the actual risk reserve.

scholarly journals Last Exit Before an Exponential Time for Spectrally Negative Lévy Processes

2009 ◽  
Vol 46 (02) ◽  
pp. 542-558 ◽  
Author(s):  
E. J. Baurdoux
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Risk Theory ◽  
Exponential Time ◽  
Main Motivation ◽  
Spectrally Negative Lévy Process ◽  
The Laplace Transform ◽  
Spectrally Negative Lévy Processes

Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to ∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).

scholarly journals On a general class of renewal risk process: analysis of the Gerber-Shiu function

2005 ◽  
Vol 37 (03) ◽  
pp. 836-856 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Process Analysis ◽  
Risk Process ◽  
Expected Discounted Penalty Function ◽  
Renewal Equations ◽  
The Laplace Transform ◽  
Discounted Penalty Function ◽  
Renewal Risk ◽  
Defective Renewal Equations ◽  
Expected Discounted Penalty

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a K n distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

scholarly journals A class of risk processes with delayed claims: ruin probability estimates under heavy tail conditions

2006 ◽  
Vol 43 (4) ◽  
pp. 916-926 ◽  
Author(s):  
Ayalvadi Ganesh ◽  
Giovanni Luca Torrisi
Keyword(s):  
Size Distribution ◽  
Ruin Probability ◽  
Heavy Tail ◽  
Claim Size ◽  
Probability Estimates ◽  
Claim Size Distribution ◽  
Risk Processes

We consider a class of risk processes with delayed claims, and we provide ruin probability estimates under heavy tail conditions on the claim size distribution.

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