Consider the optimal dividend problem for an insurance company whose uncontrolled surplus precess evolves as a spectrally negative Lvy process. We assume that dividends are paid to the shareholders according to admissible strategies whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. In this paper, we show that a threshold strategy(also called refraction strategy)forms an optimal strategy under the condition that the Lvy measure has a completely monotone density.
Let X1, X2, ··· be a sequence of dependent and heavy-tailed random variables with distributions F1, F2, ··· on (?∞, ∞), and let τ be a nonnegative integer-valued ran-dom variable independent of the sequence {Xk, k ≥ 1}. In this framework, the asymptotic behavior of the tail probabilities of the quantities Sn = n k=1 X k and S(n) = max1 ≤k≤n Sk for n > 1, and their randomized versions Sτ and S(τ) are studied. Some applications to the risk theory are presented.
In this paper,we study a general Lévy risk process with positive and negative jumps.A renewal equation and an infinite series expression are obtained for the expected discounted penalty function of this risk model.We also examine some asymptotic behaviors for the ruin probability as the initial capital tends to infinity.
