The compound Poisson risk model with multiple thresholds (original) (raw)

The compound Poisson risk model with a threshold dividend strategy

Insurance: Mathematics and Economics, 2006

In this paper we discuss a threshold dividend strategy implemented into the classical compound Poisson model. More specifically, we assume that no dividends are paid if the current surplus of the insurance company is below certain threshold level. When the surplus is above this fixed level, dividends are paid at a constant rate that does not exceed the premium rate. This model may also be viewed as the compound Poisson model with the two-step premium rate. Two integro-differential equations for the Gerber-Shiu discounted penalty function are derived and solved. When the initial surplus is below the threshold, the solution is a linear combination of the Gerber-Shiu function with no barrier and the solution of the associated homogeneous integro-differential equation. This latter function is proportional to the product of an exponential function and a compound geometric distribution function. When the initial surplus is above the threshold, the solution involves the respective Gerber-Shiu function with initial surplus lower than the threshold level. These analytic results are utilized to find 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.

On a multi-threshold compound Poisson surplus process with interest

Scandinavian Actuarial Journal, 2011

We consider a multi-threshold compound Poisson surplus process. When the initial surplus is between any two consecutive thresholds, the insurer has the option to choose the respective premium rate and interest rate. Also, the model allows for borrowing the current amount of deficit whenever the surplus falls below zero. Starting from the integro-differential equations satisfied by the GerberÁShiu function that appear in Yang et al. , we consider exponentially and phase-type(2) distributed claim sizes, in which cases we are able to transform the integro-differential equations into ordinary differential equations. As a result, we obtain explicit expressions for the GerberÁShiu function.

On a multi-threshold compound Poisson process perturbed by diffusion

Statistics & Probability Letters, 2010

We consider a multi-layer compound Poisson surplus process perturbed by diffusion and examine the behaviour of the Gerber-Shiu discounted penalty function. We derive the general solution to a certain second order integro-differential equation. This permits us to provide explicit expressions for the Gerber-Shiu function depending on the current surplus level. The advantage of our proposed approach is that if the diffusion term converges to zero, the above-mentioned explicit expressions converge to those under the classical compound Poisson model, provided that the same initial conditions apply. This is subsequently illustrated by an extended example related to the probability of ultimate ruin.

Ruin Probability Under Compound Poisson Models with Random Discount Factor

Probability in the Engineering and Informational Sciences, 2004

In this article, we consider a compound Poisson insurance risk model with a random discount factor. This model is also known as the compound filtered Poisson model. By using some stochastic analysis techniques, a convergence result for the discounted surplus process, an expression for the ruin probability, and the upper bounds for the ruin probability are obtained.

A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function

Journal of Applied Probability, 2017

We consider a Cramér–Lundberg insurance risk process with the added feature of reinsurance. If an arriving claim finds the reserve below a certain threshold γ, or if it would bring the reserve below that level, then a reinsurer pays part of the claim. Using fluctuation theory and the theory of scale functions of spectrally negative Lévy processes, we derive expressions for the Laplace transform of the time to ruin and of the joint distribution of the deficit at ruin and the surplus before ruin. We specify these results in much more detail for the threshold set-up in the case of proportional reinsurance.

The expected discounted penalty function under a risk model with stochastic income

Applied Mathematics and Computation, 2009

Defective renewal equation Erlangðn; bÞ premium distribution Expected discounted penalty function Probability of ruin Laplace transform of the time to ruin a b s t r a c t Quantities of interest in ruin theory are investigated under the general framework of the expected discounted penalty function, assuming a risk model where both premiums and claims follow compound Poisson processes. Both a defective renewal equation and an integral equation satisfied by the expected discounted penalty function are established. Some implications that these equations have on particular quantities such as the discounted deficit and the probability of ultimate ruin are illustrated. Finally, the case when premiums have Erlangðn; bÞ distribution and the distribution of the claims is arbitrary is investigated in more depth. Throughout the paper specific examples where claims and premiums have particular distributions are provided.

The discrete stationary renewal risk model and the Gerber–Shiu discounted penalty function

Insurance: Mathematics and Economics, 2004

This paper considers the stationary and the ordinary discrete renewal risk models. The main result is an expression of the Gerber-Shiu discounted penalty function in the stationary model in terms of the corresponding Gerber-Shiu function in the ordinary model. Subsequently, this relationship is considered in more detail in both the discount free case and under the compound binomial model. The latter case may be viewed as a discrete analog of the classical Poisson model. Simplifications of the general relationship are obtained, and a connection between the defective joint cumulative distribution functions of the surplus prior to ruin and the deficit at ruin in the stationary and the ordinary renewal risk models is established. Moreover, the defective probability function of the claim causing ruin is derived in the compound binomial case.

Ruin problems for a discrete time risk model with non-homogeneous conditions

Scandinavian Actuarial Journal, 2011

This paper is concerned with a non-homogeneous discrete time risk model where premiums are fixed but non-uniform, and claim amounts are independent but non-stationary. It allows one to account for the influence of inflation and interest and the effect of variability in the claims. Our main purpose is to develop an algorithm for calculating the finite time ruin probabilities and

The Effect of a threshold proportional reinsurance strategy on ruin probabilities

2009

In the context of a compound Poisson risk model, we define a threshold proportional reinsurance strategy: A retention level k1 is applied whenever the reserves are less than a determinate threshold b, and a retention level k2 is applied in the other case. We obtain the integro-differential equation for the Gerber-Shiu function (defined in Gerber and Shiu (1998)) in this model, which allows us to obtain the expressions for ruin probability and Laplace transforms of time of ruin for several distributions of the claim sizes. Finally, we present some numerical results.