On a perturbed Sparre Andersen risk model with multi-layer dividend strategy

The Gerber–Shiu discounted penalty function in a delayed renewal risk model with multi-layer dividend strategy

The Discounted Penalty Function with Multi-Layer Dividend Strategy in the Phase-Type Risk Model

In this paper, a compound Poisson risk model with time-dependent claims is studied under a multi-layer dividend strategy. A piecewise integro-differential equation for the Gerber-Shiu function is derived and solved. Asymptotic formulas of the ruin probability are obtained when the claim size distributions are heavy-tailed.

For actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (Insur Math Econ 37:324–334, 2005), by considering a general claim amount distribution.

A similar surplus analysis as that of previous chapters, but with important variations on the dependent Sparre Andersen model, is considered in this chapter. In Sect. 7.1, the joint distribution of the first interclaim time and claim size distributions is allowed to assume a different form in order to accomodate the possibility that a claim does not occur at time 0. A special case of this model involving the equilibrium distribution of subsequent (marginal) interclaim times, motivated by stationarity arguments, is also discussed. A discrete model is considered in Sect. 7.2, where special emphasis is placed on analysis using a discrete Coxian type assumption for the (marginal) interclaim times. This model generalizes the compound binomial model, a discrete analog of the classical Poisson model often used in financial applications, which is discussed here as well.

This chapter complements the material of Chap. 4 in the sense that explicit identification of requisite quantities is possible with stronger parametric assumptions about either the (marginal) distribution of the claim sizes or that of the interclaim times. Convenient parametric assumptions involve Erlang (or Coxian) distributions, due to the generality and mathematical tractability inherent in the use of such models. Sect. 5.1 directly generalizes the classical Poisson model of Chap. 3 by essentially assuming a (marginal) Erlang form for the interclaim times. A Lagrange polynomial approach together with Laplace transform arguments is used to explicitly identify all components of the Gerber-Shiu function. The independent model with an arbitrary interclaim time distribution and exponential claim sizes is then considered in Sect. 5.2. The conditioning argument of Sect. 5.2 is then extended in Sect. 5.3 to a model where the (marginal) claim size distribution is assumed to be of Coxian form. In particular, identification of the Gerber-Shiu function is shown to be possible for special choices of the penalty function, and more generally for an arbitrary penalty function of the deficit. Finally, the Coxian assumption of Sect. 5.3 is replaced by a (possibly infinite) mixture of Erlang distributions, resulting in tractable results for ruin probabilities, the deficit at ruin, and related quantities.

In this paper, we give a result on the local asymptotic behaviour of the probability of ruin in a continuous-time risk model in which the inter-claim times have an Erlang distribution and the individual claim sizes have a distribution that belongs toS(v) withv ≥ 0, but where the Lundberg exponent of the underlying risk process does not exist.

In the classical risk theory ruin probabilities were studied with the interclaim time d.f. K(t) = l-e−t. In Sparre Andersen (1957) the theory was extended to other interclaim-time distributions but doing so one has to make difference between an ordinary Sparre Andersen model and a stationary Sparre Andersen model.

In this paper, we consider an extension to the classical compound Poisson risk model. Historically, it has been assumed that the claim amounts and claim inter-arrival times are independent. In this contribution, a dependence structure between the claim amount and the interclaim time is introduced through a Farlie–Gumbel–Morgenstern copula. In this framework, we derive the integro-differential equation and the Laplace transform (LT) of the Gerber–Shiu discounted penalty function. An explicit expression for the LT of the discounted value of a general function of the deficit at ruin is obtained for claim amounts having an exponential distribution.

On the compound Poisson risk model with dependence based on a generalized Farlie–Gumbel–Morgenstern copula

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a Kn 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.

The Maximum Surplus Before Ruin for Dependent Risk Models Through Farlie-Gumbel-Morgenstern Copula

We derive formulae for the moments of the time of ruin in both ordinary and modified Sparre Andersen risk models without specifying either the inter-claim time distribution or the individual claim amount distribution. We illustrate the application of our results in the special case of exponentially distributed claims, as well as for the following ordinary models: the classical risk model, phase-type(2) risk models, and the Erlang( $\mathscr{n}$ ) risk model. We also show how the key quantities for modified models can be found.

Ruin probabilities and penalty functions with stochastic rates of interest

Analysis of the discounted sum of ascending ladder heights