A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium

In a general Sparre Andersen risk model with surplus-dependent premium income, the generalization of Gerber-Shiu function proposed by Cheung et al. (2010a) is studied. A general expression for such Gerber-Shiu function is derived, and it is shown that its determination reduces to the evaluation of a transition function which is independent of the penalty function. Properties of and explicit expressions for such a transition function are derived when the surplus process is subject to (i) constant premium; (ii) a threshold dividend strategy; or (iii) credit interest. Extension of the approach is discussed for an absolute ruin model with debit interest. © 2011 Elsevier B.V.postprin

On a class of stochastic models with two-sided jumps

In this paper a stochastic process involving two-sided jumps and a continuous downward drift is studied. In the context of ruin theory, the model can be interpreted as the surplus process of a business enterprise which is subject to constant expense rate over time along with random gains and losses. On the other hand, such a stochastic process can also be viewed as a queueing system with instantaneous work removals (or negative customers). The key quantity of our interest pertaining to the above model is (a variant of) the Gerber-Shiu expected discounted penalty function (Gerber and Shiu in N. Am. Actuar. J. 2(1):48-72, 1998) from ruin theory context. With the distributions of the jump sizes and their inter-arrival times left arbitrary, the general structure of the Gerber-Shiu function is studied via an underlying ladder height structure and the use of defective renewal equations. The components involved in the defective renewal equations are explicitly identified when the upward jumps follow a combination of exponentials. Applications of the Gerber-Shiu function are illustrated in finding (i) the Laplace transforms of the time of ruin, the time of recovery and the duration of first negative surplus in the ruin context; (ii) the joint Laplace transform of the busy period and the subsequent idle period in the queueing context; and (iii) the expected total discounted reward for a continuous payment stream payable during idle periods in a queue. © 2011 The Author(s).published_or_final_versionSpringer Open Choice, 21 Feb 201

On a Gerber–Shiu type function and its applications in a dual semi-Markovian risk model

In this paper, we consider a dual risk process which can be used to model the surplus of a business that invests money constantly and earns gains randomly in both time and amount. The occurrences of the gains and their amounts are assumed follow a semi-Markovian structure (e.g. Reinhard (1984)). We analyze a quantity resembling the Gerber-Shiu expected discounted penalty function (Gerber and Shiu (1998)) that incorporates random variables defined before and after the time of ruin, such as the minimum surplus level before ruin and the time of the first gain after ruin. General properties of the function are studied, and some exact results are derived upon exponential distributional assumptions on either the inter-arrival times or the gain amounts. Applications in a perpetual insurance and the last inter-arrival time containing the time of ruin are given along with some numerical examples.postprin

A unified analysis of claim costs up to ruin in a Markovian arrival risk model

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On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes

In this paper, a dependent Sparre Andersen risk process in which the joint density of the interclaim time and the resulting claim severity satisfies the factorization as in Willmot and Woo (2012) is considered. We study a generalization of the Gerber-Shiu function (i) whose penalty function further depends on the surplus level immediately after the second last claim before ruin (Cheung et al. (2010a)); and (ii) which involves the moments of the discounted aggregate claim costs until ruin. The generalized discounted density with a moment-based component proposed in Cheung (2013) plays a key role in deriving recursive defective renewal equations. We pay special attention to the case where the marginal distribution of the interclaim times is Coxian, and the required components in the recursion are obtained. A reverse type of dependency structure where the claim severities follow a combination of exponentials is also briefly discussed, and this leads to a nice explicit expression for the expected discounted aggregate claims until ruin. Our results are applied to generate some numerical examples involving (i) the covariance of the time of ruin and the discounted aggregate claims until ruin; and linebreak (ii) the expectation, variance and third central moment of the discounted aggregate claims until ruin.postprin

On a Risk Model with Surplus-dependent Premium and Tax Rates

In this paper, the compound Poisson risk model with surplus-dependent premium rate is analyzed in the taxation system proposed by Albrecher and Hipp (Blätter der DGVFM 28(1):13-28, 2007). In the compound Poisson risk model, Albrecher and Hipp (Blätter der DGVFM 28(1):13-28, 2007) showed that a simple relationship between the ruin probabilities in the risk model with and without tax exists. This so-called tax identity was later generalized to a surplus-dependent tax rate by Albrecher et al. (Insur Math Econ 44(2):304-306, 2009). The present paper further generalizes these results to the Gerber-Shiu function with a generalized penalty function involving the maximum surplus prior to ruin. We show that this generalized Gerber-Shiu function in the risk model with tax is closely related to the 'original' Gerber-Shiu function in the risk model without tax defined in a dividend barrier framework. The moments of the discounted tax payments before ruin and the optimal threshold level for the tax authority to start collecting tax payments are also examined. © 2010 The Author(s).published_or_final_versionSpringer Open Choice, 21 Feb 201

On the expected discounted dividends in the Cramér-Lundberg risk model with more frequent ruin monitoring than dividend decisions

In this paper, we further extend the insurance risk model in Albrecher et al. (2011b), who proposed to only intervene in the compound Poisson risk process at the discrete time points ${L_k}_{k=0}^infty$ where the event of ruin is checked and dividend decisions are made. In practice, an insurance company typically balances its books (and monitors its solvency) more frequently than deciding on dividend payments. This motivates us to propose a generalization in which ruin is monitored at ${L_k}_{k=0}^infty$ whereas dividend decisions are only made at ${L_{jk}}_{k=0}^infty$ for some positive integer $j$. Assuming that the intervals between the time points ${L_k}_{k=0}^infty$ are Erlang($n$) distributed, the Erlangization technique (e.g. Asmussen et al. (2002)) allows us to model the more realistic situation with the books balanced e.g. monthly and dividend decisions made e.g. quarterly or semi-annually. Under a dividend barrier strategy with the above randomized interventions, we derive the expected discounted dividends paid until ruin. Numerical examples about dividend maximization with respect to the barrier $b$ and/or the value of $j$ are given.postprin

On a bivariate risk process with a dividend barrier strategy

In this paper, we study a continuous-time bivariate risk process in which each individual line of business implements a dividend barrier strategy. The insurance portfolios of the two insurers are correlated as they are subject to common shocks which induce dependent claims. To analyze the expected discounted dividends until the joint ruin time of the bivariate process (i.e. exit from the positive quadrant), we propose a discrete-time counterpart of the model and apply a bivariate extension of the Dickson-Waters discretization (Dickson and Waters (1991)) with the use of a bivariate Panjer type recursion (Walhin and Paris (2000)). Detailed numerical examples under different dependencies via common shocks, copulas and proportional reinsurance are discussed, and applications to optimal problems in reinsurance, capital allocation and dividends are given. It is also illustrated that the optimal pair of dividend barriers maximizing the dividend function is dependent on the initial surplus levels. A modified type of dividend barrier strategy is proposed towards the end.postprin

The Markov Additive risk process under an Erlangized dividend barrier strategy

In this paper, we consider a Markov additive insurance risk process under a randomized dividend strategy in the spirit of Albrecher et al. (2011). Decisions on whether to pay dividends are only made at a sequence of dividend decision time points whose intervals are Erlang($n$) distributed. At a dividend decision time, if the surplus level is larger than a predetermined dividend barrier, then the excess is paid as a dividend as long as ruin has not occurred. In contrast to Albrecher et al. (2011), it is assumed that the event of ruin is monitored continuously (Avanzi et al. (2013) and Zhang (2014)), i.e. the surplus process is stopped immediately once it drops below zero. The quantities of our interest include the Gerber-Shiu expected discounted penalty function and the expected present value of dividends paid until ruin. Solutions are derived with the use of Markov renewal equations. Numerical examples are given, and the optimal dividend barrier is identified in some cases.postprin

On the time value of Parisian ruin in (dual) renewal risk processes with exponential jumps

This paper studies the Parisian ruin problem first proposed by Dassios and Wu (2008a,b), where the Parisian ruin time is defined to be the first time when the surplus process has stayed below zero continuously for a pre-specified time length $d$. Both the insurance risk process and the dual model will be considered under exponential distributional assumption on the jump sizes while keeping the inter-arrival times arbitrary. In these two models, the Laplace transform of the Parisian ruin time is derived by extending the excursion techniques in Dassios and Wu (2008a) and taking advantage of the memoryless property exponential distributions. Our results are represented in integral forms, which are expressed in terms of the (joint) densities of various ruin-related quantities that are available in the literature or obtainable using the Lagrange's expansion theorem. As a by-product, we also provide the joint distribution of the numbers of periods of negative surplus that are of duration more than $d$ and less than $d$, which can be obtained using some of our intermediate results. The case where the Parisian delay period $d$ is replaced by a random time is also discussed, and it is applied to find the Laplace transform of the occupation time when the surplus is negative. Numerical illustrations concerning an Erlang(2) insurance risk model are given at the end.postprin