The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Lévy process and the solution of Gerber and Shiu [Astin

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 amongst all admissible ones takes the form of a barrier strategy.

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muller (2005) and Avram et al. (2006) which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically we build on recent work in the actuarial literature concerning calculations for the n-th moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than existing literature in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér-Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and for the case of the n-th moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Lévy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations.

Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.

For spectrally negative Markov Additive Processes (MAPs) we generalize classical fluctuation identities developed in Zolotarev (1964), Takács (1967),

Motivated by a classical control problem from actuarial mathematics, we study smoothness and convexity properties of q-scale functions for spectrally negative Lévy processes. Continuing from the very recent work of [2] and [24] we strengthen their collective conclusions by showing, amongst other results, that whenever the Lévy measure has a non-increasing density which is log convex then for q> 0 the scale function W (q) is convex on some half line (a ∗ , ∞) where a ∗ is the largest value at which W (q)′ attains its global minimum. As a consequence we deduce that de Finetti’s classical actuarial control problem is solved by a barrier strategy where the barrier is positioned at height a ∗.

In this paper we overview the pricing of several so-called exotic options in the nowdays quite popular exponential Lévy models.

Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holder's claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating an optimal stopping problem assocaited with Dynkin games. In this short text we give two examples of perpetual Israeli options where the solutions are explicit.

Abstract. According to a theorem of S. Schumacher and T. Brox, for a diffusion X in a Brownian environment it holds that (Xt − blog t)/log 2 t → 0 in probability, as t → ∞, where b · is a stochastic process having an explicit description and depending only on the environment. In the first part of this paper we compute the distribution of the sign changes for b on an interval [1, x] and study some of the consequences of the computation; in particular we get the probability of b keeping the same sign on that interval. These results have been announced in 1999 in a non-rigorous paper by P. Le Doussal, C. Monthus, and D. Fisher and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. In the second part we consider the case that the environment is a spectrally one sided stable process and derive results describing the features of the environment that matter for the study of the process b. In particular we derive the distribution of b1. 1.