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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

THE TIME VALUE OF R UIN IN A SPARRE ANDERSEN MODEL   (Citations: 56)
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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 integro-differential equation and that this equation can be solved in terms of Laplace transforms, extending a result found in Lin (2003). As a consequence, a closed-form 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 closed-form 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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    • ...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 integro-differential 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 Zhanget 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 Gerber-Shiu 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 Albrecheret al. An Algebraic Operator Approach to the Analysis of Gerber-Shiu Function...

    • ...Li and Garrido (2004) and Gerber and Shiu (2005) were the first to investigate the Gerber-Shiu 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 Albrecheret al. An Algebraic Approach to the Analysis of Gerber-Shiu 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 Yuanet 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 Chiragievet al. Multivariate Pareto portfolios: TCE-based capital allocation and divid...

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