In this paper, we discuss a generalization of the classical compound Poisson model with claim sizes following a compound distribution. As applications, we consider models involving zero-truncated geometric, zero-truncated negative-binomial and zero-truncated binomial batch-claim arrivals. We also provide some ruin-related quantities under the resulting risk models. Finally, through numerical examples, we visualize the behavior of these quantities.

In this paper, we model the surplus process as a compound Poisson process perturbed by diffusion and allow the insurer to ask its customers for input to minimize the distance from some prescribed target path and the total discounted cost on a fixed interval. The problem is reduced to a version of a linear quadratic regulator under jump-diffusion processes. It is treated using three methods: dynamic programming, completion of square and the stochastic maximum principle. The analytic solutions to the optimal control and the corresponding optimal value function are obtained.

We consider a two-dimensional optimal dividend problem in the context of two insurance companies with compound Poisson surplus processes, who collaborate by paying each other's deficit when possible. We study the stochastic control problem of maximizing the weighted sum of expected discounted dividend payments (among all admissible dividend strategies) until ruin of both companies, by extending results of univariate optimal control theory. In the case that the dividends paid by the two companies are equally weighted, the value function of this problem compares favorably with the one of merging the two companies completely. We identify the optimal value function as the smallest viscosity supersolution of the respective Hamilton–Jacobi–Bellman equation and provide an iterative approach to approximate it numerically. Curve strategies are identified as the natural analogue of barrier strategies in this two-dimensional context. A numerical example is given for which such a curve strategy is indeed optimal among all admissible dividend strategies, and for which this collaboration mechanism also outperforms the suitably weighted optimal dividend strategies of the two stand-alone companies.

We consider the optimal proportional reinsurance problem for an insurer with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Using the technique of stochastic linear–quadratic control theory and the Hamilton–Jacobi–Bellman equation, we derive the explicit expressions for the optimal reinsurance strategies and value function, and present the verification theorem within the framework of the viscosity solution. Furthermore, we extend the results in the linear–quadratic setting to the mean–variance problem, and obtain an efficient strategy and frontier. Some numerical examples are given to show the impact of model parameters on the efficient frontier.

Superposition is a mapping on point configurations that sends the n-tuple into the n-point configuration , counted with multiplicity. It is an additive set operation such that the superposition of a k-point configuration in is a kn-point configuration in . A Poisson superposition process is the superposition in of a Poisson process in the space of finite-length -valued sequences. From properties of Poisson processes as well as some algebraic properties of formal power series, we obtain an explicit expression for the Janossy measure of Poisson superposition processes, and we study their law under domain restriction. Examples of well-known Poisson superposition processes include compound Poisson, negative binomial, and permanental (boson) processes.

In this paper, we compare the error in several approximation methods for the cumulative aggregate claim distribution customarily used in the collective model of insurance theory. In this model, it is usually supposed that a portfolio is at risk for a time period of length t. The occurrences of the claims are governed by a Poisson process of intensity μ so that the number of claims in [0,t] is a Poisson random variable with parameter λ = μ t. Each single claim is an independent replication of the random variable X, representing the claim severity. The aggregate claim or total claim amount process in [0,t] is represented by the random sum of N independent replications of X, whose cumulative distribution function (cdf) is the object of study. Due to its computational complexity, several approximation methods for this cdf have been proposed. In this paper, we consider 15 approximations put forward in the literature that only use information on the lower order moments of the involved distributions. For each approximation, we consider the difference between the true distribution and the approximating one and we propose to use expansions of this difference related to Edgeworth series to measure their accuracy as λ = μ t diverges to infinity. Using these expansions, several statements concerning the quality of approximations for the distribution of the aggregate claim process can find theoretical support. Other statements can be disproved on the same grounds. Finally, we investigate numerically the accuracy of the proposed formulas.

In this paper we study the distribution of the location, at time t, of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of U, V, and W are absolutely continuous. The velocities are v = +1 upwards, v = -1 downwards, and v = 0 during idle periods. Let Y+(t), Y−(t), and Y0(t) denote the total time in (0, t) of movements upwards, downwards, and no movements, respectively. The exact distribution of Y+(t) is derived. We also obtain the probability law of X(t) = Y+(t) - Y−(t), which describes the particle's location at time t. Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).

In this paper we investigate the potential of Lévy copulas as a tool for modelling dependence between compound Poisson processes and their applications in insurance. We analyse characteristics regarding the dependence in frequency and dependence in severity allowed by various Lévy copula models. Through the introduction of new Lévy copulas and comparison with the Clayton Lévy copula, we show that Lévy copulas allow for a great range of dependence structures.

Procedures for analysing the fit of Lévy copula models are illustrated by fitting a number of Lévy copulas to a set of real data from Swiss workers compensation insurance. How to assess the fit of these models with respect to the dependence structure exhibited by the dataset is also discussed.

Finally, we provide a decomposition of the trivariate compound Poisson process and discuss how trivariate Lévy copulas model dependence in this multivariate setting.

We consider an insurance company whose surplus is governed by a jump diffusion risk process. The insurance company can purchase proportional reinsurance for claims and invest its surplus in a risk-free asset and a risky asset whose return follows a jump diffusion process. Our main goal is to find an optimal investment and proportional reinsurance policy which maximizes the expected exponential utility of the terminal wealth. By solving the corresponding Hamilton–Jacobi–Bellman equation, closed-form solutions for the value function as well as the optimal investment and proportional reinsurance policy are obtained. We also discuss the effects of parameters on the optimal investment and proportional reinsurance policy by numerical calculations.

In this paper, we investigate the computation of the moments of the compound Poisson sums with discounted claims when introducing dependence between the interclaim time and the subsequent claim size. The dependence structure between the two random variables is defined by a Farlie-Gumbel-Morgenstern copula. Assuming that the claim distribution has finite moments, we give expressions for the first and the second moments and then we obtain a general formula for any mth order moment. The results are illustrated with applications to premium calculation and approximations based on moment matching methods.

Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function Fs (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂nFs (x; t) ∂ xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.

Assume that the surplus of an insurer follows a compound Poisson surplus process. When the surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay the debts from her premium income. The negative surplus may return to a positive level. However, when the negative surplus is below a certain critical level, the surplus is no longer able to be positive. Absolute ruin occurs at this moment. In this paper, we study absolute ruin questions by defining an expected discounted penalty function at absolute ruin. The function includes the absolute ruin probability, the Laplace transform of the time to absolute ruin, the deficit at absolute ruin, the surplus just before absolute ruin, and many other quantities related to absolute ruin. First, we derive a system of integro-differential equations satisfied by the function and obtain a defective renewal equation that links the integro-differential equations in the system. Second, we show that when the initial surplus goes to infinity, the absolute ruin probability and the classical ruin probability are asymptotically equal for heavy-tailed claims while the ratio of the absolute ruin probability to the classical ruin probability goes to a positive constant that is less than one for light-tailed claims. Finally, we give explicit expressions for the function for exponential claims.

For a general class of risk models, the dividends-penalty identity is derived by probabilistic reasoning. This identity is the key for understanding and determining the optimal dividend barrier, which maximizes the difference between the expected present value of all dividends until ruin and the expected discounted value of a penalty at ruin (which is typically a function of the deficit at ruin). As an illustration, the optimal barrier is calculated in two classical models, for different penalty functions and a variety of parameter values.

Let denote the class of local subexponential distributions and F∗ν the ν-fold convolution of distribution F, where ν belongs to one of the following three cases: ν is a random variable taking only a finite number of values, in particular ν ≡ n for some n ≥ 2; ν is a Poisson random variable; or ν is a geometric random variable. Along the lines of Embrechts, Goldie, and Veraverbeke (1979), the following assertion is proved under certain conditions: This result is applied to the infinitely divisible laws and some new results are established. The results obtained extend the corresponding findings of Asmussen, Foss, and Korshunov (2003).

We consider a finite dam under the policy, where the input of water is formed by a Wiener process subject to random jumps arriving according to a Poisson process. The long-run average cost per unit time is obtained after assigning costs to the changes of release rate, a reward to each unit of output, and a penalty that is a function of the level of water in the reservoir.

We consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim inter-arrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the Kn family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

Burn-in is a widely used engineering procedure useful for eliminating ‘weak’ items and consequently improving the quality of remaining items. The quality of items can be measured via various performance characteristics. In the present paper we develop new performance criteria for the burn-in method. Our criteria not only take into account the reliability of an item, they also incorporate covariates.