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

The Gerber–Shiu discounted penalty function in the delayed renewal risk process with random income

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

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

of thesis entitled STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES Submitted by CHEN YIQING for the degree of Doctor of Philosophy at The University of Hong Kong in August 2009 In recent years, insurance risk studies in the presence of dependence structure and heavy-tailed claims are popular topics in actuarial science. In this thesis, we consider the renewal risk model with certain dependence structures, namely negative dependence and quasi-asymptotic independence. We also assume that claim sizes follow a heavy-tailed distribution, in particular, a subexponential distribution. We focus on studying the impact of heavy tails and dependence structures on ruin probabilities and the tail probabilities of aggregate claims. The asymptotic behavior of the ultimate ruin probability in the renewal risk model was studied in the presence of negatively dependent claim sizes and a constant force of interest. A purely probabilistic approach was employed to establish a simple asymptotic formula for the ruin probability. This result coincides with some known results in the recent literature, and discovers a phenomenon that heavy tails can offset the impact of the dependence structure of claim sizes on the ruin probability. A two-dimensional renewal risk model was considered in which the insurance company consists of two dependent classes of business. The dependence between the two classes comes from the fact that their claims arrive according to a common renewal counting process. For this model, several ruin probabilities were defined and simple asymptotic formulae for these ruin probabilities were derived. In particular, the obtained asymptotic formulae hold uniformly for all time horizons. This uniformity greatly enhances the theoretical value of the results. The next problem was to study precise estimates of the large deviation probabilities of aggregate claims with sizes which are negatively dependent. Specially, the problem for the so-called quasi renewal risk model was investigated in which inter-arrival times are also negatively dependent. A new dependence structure was considered and was termed “quasi-asymptotic independence” structure, which is more general and verifiable than some commonlyused dependence structures. The proposed dependence structure covers a wide range of positive and negative dependence. For the case of Pareto-like tails, the tail probability of the sum was proved to be asymptotically equivalent to the sum of tail probabilities. This confirms again that heavy tails offset the impact of dependence between summands. This result was then extended to randomly weighted sums and random sums. All the obtained results have obvious applications to concrete actuarial problems. In the last part of the thesis, each primary random variable representing the net loss during a period was associated with a random weight or a stochastic discount factor. Therefore, the weighted sum represents the stochastic present value of aggregate losses. Suppose that the primary random variables are long tailed, the weights are positive and bounded, and the primary random variables and their weights are independent. The tail probability of the maximum of the weighted sums was shown to be asymptotically equivalent to that of the last weighted sum. STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES

The Gerber-Shiu discounted penalty function is considered for a class of delayed renewal risk processes. In (Willmot 2004), special cases of the model include the stationary renewal risk model and the situation where the time until the first claim is exponentially distributed. In this paper, we consider a class of delayed and perturbed risk model with dependence between interclaim arrivals and claim sizes. The integro-differential equations for the Gerber-Shiu discounted penalty functions are derived.

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.

this paper discusses ruin problems in the renewal risk model with interest force. By using inductive method, the recursive expressions of the distribution of the ruin probability, the distribution of maximum surplus before the ruin and distribution of minimum surplus before the ruin are obtained, and the corresponding integral equation for the distributions are obtained. Keywords-risk model; Interest force; renewal risk model; ruin probability; maximum surplus; minimum surplus

Let \(R(t) = u + ct - \sum\limits_{i = 1}^{N(t)} {X_i ,t \geqslant 0} \) be the renewal risk model, with FX(x) being the distribution function of the claim amount X. Let ψ(u) be the ruin probability with initial surplus u. Under the condition of FX(x) ∈ S*(γ, γ ⩾ 0, by the geometric sum method, we derive the local asymptotic behavior for ψ(u, u + z] for every 0 < z < ∞. On one hand, the asymptotic behavior of ψ(u) can be derived from the result obtained. On the other hand, the result of this paper can be applied to the insurance risk management of an insurance company.

Kluppelberg and Stadtmuller (1998) proved a precise asymptotic formula for the ruin probability of the classical interest force and regularly varying tailed claims when the initial capital u tends to infinity . This paper extends their results in several aspects as follows: First, the risk models are the Conditional Poisson and Non-Qici Poisson Process respectively; Second, the ruin probability is replaced by the finite time ruin probability within time T and the claimsize is of Subexponential family which is more wide then that of regularly varying's family. At last, the diffusion term of Brownian Motion is considered. construct et ψ(u;T) be the finite time ruin probability in the renewal risk model, where u is the initial capital of the company and $T$ denotes some given time bound. Under the assumption that the distribution of the claim size belongs to the Extended regular variation class, this paper obtains an asymptotic formula for ψ(u;T). This result improves the related works in the recent literature.

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.

Constant dividend barrier in a risk model with interclaim-dependent claim sizes

In this paper the probability of ruin is investigated under the influence of a premium rate which varies according to the intensity of claims, and the occurrence of claims is described by a Cox process in the considered risk model. The idea is originally enlightened by Jasiulewicz (2001). We make a slight modification on the model and a generalization on the intensity process. By a “backward differential argument” and the Markov property of the intensity process we strictly derive the integral equation satisfied by the probability of ruin. Further, we solve the equation when the intensity process is a homogeneous n-state Markov process by Laplace transforms. At the end of the paper, an example is given.

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.

We study the expected time to ruin in a risk process in which dividends are paid when the surplus is above the barrier. We consider the case in which the dividend rate is smaller than the premium rate. We obtain results for the classical compound Poisson risk process with phase-type claim size. When the ruin probability is 1, we derive the expected time to ruin and the expected dividends paid. When the ruin probability is less than 1, these quantities are derived conditioning on the event that ruin occurs.

Consider the probability of random time ruin in the renewal risk model with the general nonnegative and non decreasing premium process and constant interest rate. We obtain a uniform asymptotic formula for random time τ and subexponential distribution.