ABSTRACT. We study the optimal dividend problem in the dual model where dividend payments can only be made at the jump times of an independent Poisson process. In this context, Avanzi et al. [6] solved the case with i.i.d. hyperexponential jumps; they showed the optimality of a (periodic) barrier strategy where dividends are paid at dividend-decision times if and only if the surplus is above some level. In this paper, we generalize the results for a general spectrally positive Lévy process with additional terminal payoff/penalty at ruin, and also solve the case with classical bail-outs so that the surplus is restricted to be nonnegative.The optimal strategies as well as the value functions are concisely written in terms of the scale function.Numerical results are also given.AMS 2010 Subject Classifications: 60G51, 93E20, 91B30 JEL Classifications: C44, C61, G24, G32, G35

In the last few years there has been renewed interest in the classical control problem of de Finetti (1957) for the case where the underlying source of randomness is a spectrally negative Lévy process. In particular, a significant step forward was made by Loeffen (2008), who showed that a natural and very general condition on the underlying Lévy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Lévy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem, but with the restriction that control strategies are absolutely continuous with respect to the Lebesgue measure. This problem has been considered by Asmussen and Taksar (1997), Jeanblanc-Picqué andShiryaev (1995), and Boguslavskaya (2006) in the diffusive case, and Gerber and Shiu (2006) for the case of a Cramér-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Lévy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.

In this paper we introduce a new class of Lévy processes which we call hypergeometricstable Lévy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the Lévy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems.1 for all t ≥ 0 and θ ∈ R.According to Bertoin (2), the process Z is transient for α < d, that is lim t→∞ Z t = ∞ a.s., and it oscillates otherwise, i.e. for α ∈ [1, 2) and d = 1, we have lim supWhen d ≥ 2, we have that single points are polar, i.e. for every x, z ∈ R d P x (Z t = z for some t > 0) = 0.In the one-dimensional case, points are polar for α ∈ (0, 1] and when α ∈ (1, 2) the process Z makes infinitely many jumps across a point, say z, before the first hitting time of z (see for instance Proposition VIII.8 in (2)). One of the main properties of the process Z is that it satisfies the scaling property with index α, i.e. for every b > 0

ABSTRACT. We study a combination of the refracted and reflected Lévy processes. Given a spectrally negative Lévy process and two boundaries, it is reflected at the lower boundary while, whenever it is above the upper boundary, a linear drift at a constant rate is subtracted from the increments of the process. Using the scale functions, we compute the resolvent measure, the Laplace transform of the occupation times as well as other fluctuation identities that will be useful in applied probability including insurance, queues, and inventory management.

A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu [12]. Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used.