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Keywords
(9)
Collective Risk Model
Deficit at Ruin
Divided Difference
Integrodifferen...
Laplace Transform
Probability Density
Random Variable
Satisfiability
sparre andersen model
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(2)
Moments of the surplus before ruin and the defecit at ruin in the Erlang(2) risk process
On the time to ruin for Erlang(2) risk processes
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THE TIME VALUE OF R UIN IN A SPARRE ANDERSEN MODEL
THE TIME VALUE OF R UIN IN A SPARRE ANDERSEN MODEL,Hans U. Gerber,Elias S. W. Shiu
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THE TIME VALUE OF R UIN IN A SPARRE ANDERSEN MODEL
(
Citations: 56
)
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Hans U. Gerber
,
Elias S. W. Shiu
This paper considers a Sparre Andersen
collective risk model
in which the distribution of the interclaim time is that of a sum of n independent exponential random variables; thus, the Erlang(n) model is a special case. The analysis is focused on the function (u), the expected discounted penalty at ruin, with u being the initial surplus. The penalty may depend on the
deficit at ruin
and possibly also on the surplus immediately before ruin. It is shown that the function (u) satisfies a certain
integrodifferential equation
and that this equation can be solved in terms of Laplace transforms, extending a result found in Lin (2003). As a consequence, a closedform expression is obtained for the discounted joint
probability density
of the
deficit at ruin
and the surplus just before ruin, if the initial surplus is zero. For this formula and other results, the roots of Lundberg's fundamental equation in the right half of the complex plane play a central role. Also, it is shown that (u) satisfies Li's (2003) renewal equation. Under the assumption that the penalty depends only on the
deficit at ruin
and that the individual claim amount density is a combination of exponential densities, a closedform expression for (u) is derived. In this context, known results of the Cauchy matrix are useful. Surprisingly, certain results are best expressed in terms of divided differences, a topic deleted from the actuarial examinations at the end of last century.
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Citation Context
(9)
...Some recent contributions to Sparre Anderson model are Dickson and Hipp [1, 2], Cheng and Tang [3], Dickson and Drekic [4], Gerber and Shiu [
5
], Li and Garrido [6, 7], Li and Lu [8], Li and Dickson [9], Sun and Liang [10], as well as the references therein...
...For a function h(s), let h[s, r1 ,r 2 ,...,r n] be its divided differences with respect to r1 ,r 2 ,..., rn. See Gerber and Shiu [
5
] for the definition of divided differences...
...Before deriving the maximum surplus distribution, we consider, in terminology of Gerber and Shiu [
5
, 20], the Lundberg’s fundamental equation for our model...
...Remark 3.1 With σ = 0, equation (3.3) yields equation (1.8) of Gerber and Shiu [
5
] and equation (3.1) of Dickson and Hipp [2]...
...The solution to this type of homogenous integrodifferential equation, which has been studied by Dickson and Hipp [2], Gerber and Shiu [
5
], Li and Garrido [7], is uniquely determined by the 2n boundary conditions, and can be solved by Laplace transforms...
...Note that r(s) is a polynomial of degree 2n. From the definition of divided differences (see Gerber and Shiu [
5
]), we see that r[s, ρ1 ,ρ 2 ,...,ρ n] is a polynomial of degree n and the leading coefficient is equal to the coefficient of s 2n in r(s), which is γn := (−1) n σ...
Zhen Zhong Zhang
,
et al.
The maximum surplus distribution before ruin in an Erlang( n ) risk pr...
...Li and Garrido (2004) and
Gerber and Shiu (2005)
were the first to investigate the GerberShiu function in renewal models...
...tion (3.8) of Cheng and Tang (2003) and similar examples with n = 2 from Li and Garrido (2004, 2005b) and
Gerber and Shiu (2005)
...
Hansjorg Albrecher
,
et al.
An Algebraic Operator Approach to the Analysis of GerberShiu Function...
...Li and Garrido (2004) and
Gerber and Shiu (2005)
were the first to investigate the GerberShiu function in renewal models...
...tion (3.8) of Cheng and Tang (2003) and similar examples with n = 2 from Li and Garrido (2004), Li and Garrido (2005b),
Gerber and Shiu (2005)
...
Hansjorg Albrecher
,
et al.
An Algebraic Approach to the Analysis of GerberShiu Functions
...This function has also been employed in the Sparre Anderson risk model (see Li and Garrido
^{[}
^{15}
^{]}
, Gerber and Shiu
^{[}
^{12}
^{]}
, and the references therein)...
Haili Yuan
,
et al.
The Compound Poisson Risk Model with Interest and a Threshold Strategy
...The representation of density function of
S
by the divided differences was used for different purposes in Gerber & Shiu (
2005
)...
Arthur Chiragiev
,
et al.
Multivariate Pareto portfolios: TCEbased capital allocation and divid...
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Stochastic Processes for Insurance and Finance
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Citations: 201
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T. Rolski
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H. Schmidli
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J. L. Teugels
Published in 1999.
O NT HEMOMENTS OF THE TIME OF RUIN WITH APPLICATIONS TO PHASETYPE CLAIMS
(
Citations: 1
)
Steve Drekic
,
Gordon E. Willmot
Ruin Probabillities and Deficit for the Renewal Risk Model with PhaseType Interarrival Times
(
Citations: 16
)
F. AVRAM
,
M. USÁBEL
Journal:
Astin Bulletin  ASTIN BULL
, vol. 34, no. 2, pp. 315332, 2004
Sort by:
Citations
(56)
GerberShiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times
(
Citations: 1
)
Zhimin Zhang
,
Hu Yang
Journal:
Journal of Computational and Applied Mathematics  J COMPUT APPL MATH
, vol. 235, no. 5, pp. 11891204, 2011
The maximum surplus distribution before ruin in an Erlang( n ) risk process perturbed by diffusion
Zhen Zhong Zhang
,
Jie Zhong Zou
,
Yuan Yuan Liu
Journal:
Acta Mathematica Sinicaenglish Series  ACTA MATH SINENGLISH SERIES
, vol. 27, no. 9, pp. 18691880, 2011
Structural properties of Gerber–Shiu functions in dependent Sparre Andersen models
(
Citations: 12
)
Eric C. K. Cheung
,
David Landriault
,
Gordon E. Willmot
,
JaeKyung Woo
Journal:
Insurance Mathematics & Economics  INSUR MATH ECON
, vol. 46, no. 1, pp. 117126, 2010
Asymptotic aspects of the Gerber–Shiu function in the renewal risk model using Wiener–Hopf factorization and convolution equivalence
(
Citations: 2
)
Qihe Tang
,
Li Wei
Journal:
Insurance Mathematics & Economics  INSUR MATH ECON
, vol. 46, no. 1, pp. 1931, 2010
On the Gerber–Shiu function and change of measure
(
Citations: 2
)
Hanspeter Schmidli
Journal:
Insurance Mathematics & Economics  INSUR MATH ECON
, vol. 46, no. 1, pp. 311, 2010