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

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.

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

Through Laplace transforms, we study the extremes of a continuous-time Markov-additive process with one-sided jumps and a finite background Markovian state-space, jointly with the epoch at which the extreme is ‘attained’. For this, we investigate discrete-time Markov-additive processes and use an embedding to relate these to a continuous-time setting. The resulting Laplace transform is given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. Then we turn to tandem fluid networks, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product

Consider a Levy process with nite intensity positive jumps of the phase-type and arbitrary negative jumps. Assume that the process either is killed at a constant rate or drifts to 1. We show that the distribution of the overall maximum of this process is also of phase-type, and nd the distribution of this random variable. Previous results (hyperexponential positive jumps) are obtained as a particular case.

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

In this survey, we show that various stochastic optimization problems arising in option theory, in dynamical allocation problems, and in the microeconomic theory of intertemporal consumption choice can all be reduced to the same problem of representing a given stochastic process in terms of running maxima of another process. We describe recent results of Bank and El Karoui (2002) on the general stochastic representation problem, derive results in closed form for Lévy processes and diffusions, present an algorithm for explicit computations, and discuss some applications. AMS 2000 subject classification. 60G07, 60G40, 60H25, 91B16, 91B28. Key words and phrases. American options, Gittins index, multi–armed bandits, optimal

We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes: • Phase-type upward jumps: we find the joint distribution of the supremum and the epoch at which it is ‘attained ’ if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. • Perturbed risk models: we establish general properties, and obtain explicit fluctuation identities in case the Lévy process is spectrally positive. • Asymptotics for Lévy processes: we study the tail distribution of the supremum under different assumptions on the tail of the Lévy measure. Key words: first factorization identity, Lévy processes, perturbed risk model, phase-type jumps, ruin probability. 1

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique, which can be used to derive various further identities.

ABSTRACT. We consider a semi-Markov additive process A(·), i.e., a Markov additive process for which the sojourn times in the various states have general (rather than exponential) distributions. Letting the Lévy processes Xi(·), which describe the evolution of A(·) while the background process is in state i, be increasing, it is shown how double transforms of the type ∫ ∞ 0 e−qt E[e −αA(t)]dt can be computed. It turns out that these follow, for given α ≥ 0 and q> 0, from a system of linear equations, which has a unique positive solution. Several extensions are considered as well. 1.