In this paper, the objective is to provide sequences of improved non-increasing (non-decreasing) upper (lower) bounds for the solution of (defective) renewal equations in terms of the right-tail ...probability of a compound geometric distribution. Exponential (Lundberg type) and non-exponential type bounds are also derived. Also, under several reliability classifications, some new as well as improvements of well-known bounds are given. The results are then applied to obtain refinements of the bounds for several ruin related quantities, (such as the deficit at ruin, the joint distribution of the surplus prior to and at ruin, the mean deficit at ruin and the stop-loss premium, and the compound geometric densities). Bounds for the renewal function are also given.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
In this paper, we study the statistical estimation of the discounted density of the deficit at ruin in the classical risk model. The estimator is constructed by the two-dimensional Fourier cosine ...series expansion. It is shown that the estimator is easily computed and has fast convergence rate. Some simulation results are presented to show that the estimator performs very well when the sample size is finite.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPUK, ZAGLJ, ZRSKP
In this paper, a renewal model of risk theory is considered, and relations between tails of the equilibrium distribution of the deficit at the time of ruin and a generalization of the equilibrium ...distribution of non-ruin are obtained. Extensions to higher order equilibrium distribution and Gerber-Shiu functions are also provided. Distributional properties for Pareto and exponential claims are studied. The results are illustrated by numerical examples.
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BFBNIB, GIS, IJS, KISLJ, NUK, PNG, UL, UM, UPUK
In a Lévy insurance risk model, under the assumption that the tail of the Lévy measure is log-convex, we show that either a horizontal barrier strategy or the take-the-money-and-run strategy ...maximizes, among all admissible strategies, the dividend payments subject to an affine penalty function at ruin. As a key step for the proof, we prove that, under the aforementioned condition on the jump measure, the scale function of the spectrally negative Lévy process has a log-convex derivative.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPUK
This paper proposes a new insurance model, called More for Less, as an alternative to (re)insurance. Briefly, the company charges more premiums than necessary from the insured, but it undertakes to ...reimburse part of them if there has been no claim. Our goal is to compare this model to a (re)insurance model by examining the finite-time ruin probabilities and the expected deficit at ruin. The approach adopted here is based on simple calculations of path integrals and properties of an underlying family of Sheffer polynomials. The main motivation is to offer a different insurance coverage in today’s changing world.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
The ruin problem has long since received much attention in the literature. Under the classical compound Poisson risk model, elegant results have been obtained in the past few decades. We revisit the ...finite-time ruin probability by using the idea of cycle lemma, which was used in proving the ballot theorem. The finite-time result is then extended to infinite-time horizon by applying the weak law of large numbers. The cycle lemma also motivates us to study the claim instants retrospectively, and this idea can be used to reach the ladder height distribution on the infinite-time horizon. The new proofs in this paper link the classical finite-time and infinite-time ruin results, and give an intuitive way to understand the nature of ruin.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPUK, ZAGLJ, ZRSKP
A defective renewal equation is derived for the expected discounted penalty due at ruin,
φ
δ(u)=E
e
−δTw(U(T
−),|U(T)|)I(T<∞)|U(0)=u,
in a risk model with Erlang(
n) claim inter-arrival times. The ...approach used is similar to that of
Gerber and Shiu (1998), repeatedly integrating an
nth order integro-differential equation for
φ,
n times. The joint distribution of three random variables is obtained, the ruin time, the surplus just before ruin and the deficit at ruin.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPUK
8.
A state dependent reinsurance model Boxma, Onno; Frostig, Esther; Perry, David ...
Insurance, mathematics & economics,
05/2017, Volume:
74
Journal Article
Peer reviewed
Open access
We consider the surplus of an insurance company that employs reinsurance. The reinsurer covers part of the claims, but in return it receives a certain part of the income from premiums of the ...insurance company. In addition, the reinsurer receives some of the dividends that are withdrawn when a certain surplus level b is reached.
A special feature of our model is that both the fraction of the premium that goes to the reinsurer and the fraction of the claims covered by the reinsurer are state-dependent. We focus on five performance measures, viz., time to ruin, deficit at ruin, the dividend withdrawn until ruin, and the amount of money transferred to the reinsurer, respectively covered by the reinsurer.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPUK, ZRSKP
In this paper, we present the classical compound Poisson risk model with a threshold dividend strategy. Under such as strategy, no dividends are paid if the insurer’s surplus is below certain ...threshold level. When the surplus is above this threshold level, dividends are paid at a constant rate that does not exceed the premium rate. Two integro-differential equations for the Gerber–Shiu discounted penalty function are derived and solved. The analytic results obtained are utilized to derive the probability of ultimate ruin, the time of ruin, the distribution of the first surplus drop below the initial level, and the joint distributions and moments of the surplus immediately before ruin and the deficit at ruin. The special cases where the claim size distribution is exponential and a combination of exponentials are considered in some detail.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPUK
In this paper we generalize a partial integrodifferential equation satisfied by the finite time ruin probability in the classical Poisson risk model. The generalization also includes the bivariate ...distribution function of the time of and the deficit at ruin. We solve the partial integrodifferential equation by Laplace transforms with the help of Lagrange’s implicit function theorem. The assumption of mixed Erlang claim sizes is then shown to result in tractable computational formulas for the finite time ruin probability as well as the bivariate distribution function of the time of and the deficit at ruin. A more general partial integrodifferential equation is then briefly considered.
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