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In this paper we consider the optimal dividend problem for an insurance company whose risk process evolves as a spectrally negative Lévy process in the absence of dividend payments. The classical dividend problem for an insurance company consists in finding a dividend payment policy that maximizes the total expected discounted dividends. Related is the problem where we impose the restriction that ruin be prevented: the beneficiaries of the dividends must then keep the insurance company solvent by bail-out loans. Drawing on the fluctuation theory of spectrally negative Lévy processes we give an explicit analytical description of the optimal strategy in the set of barrier strategies and the corresponding value function, for either of the problems. Subsequently we investigate when the dividend policy that is optimal among all admissible ones takes the form of a barrier strategy.1. Introduction. In classical collective risk theory (e.g., [11]) the surplus X = {X t , t ≥ 0} of an insurance company with initial capital x is described by the Cramér-Lundberg model:

In this paper we study the joint ruin problem for two insurance companies that divide between them both claims and premia in some specified proportions (modeling two branches of the same insurance company or an insurance and re-insurance company). Modeling the risk processes of the insurance companies by Cramér-Lundberg processes we obtain the Laplace transform in space of the probability that either of the insurance companies is ruined in finite time. Subsequently, for exponentially distributed claims, we derive an explicit analytical expression for this joint ruin probability by explicitly inverting this Laplace transform. We also provide a characterization of the Laplace transform of the joint ruin time.

We prove central limit theorems (CLT) for the following problems in geometrical probability when points are generated in the [0, 1]'^c ube according to a Poisson point process with parameter n:1. The length of the /bth-nearest graph Nk^", in which each point is connected to its k-th nearest neighbor.2. The length of the Delaunay triangulation Z)" of the points. We show using the technique of dependency graphs of Baldi and Rinot that the de-pendence rajige in all these problems converges quickly to with high probability. In addition, our approach leads to more efficient sequenticd and parallel zJgorithms for this class of problems on randomly distributed points.

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Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cramér light-tail assumption on the claim size distribution.

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