In this paper we apply the recently established Wiener-Hopf Monte Carlo simulation technique for Lévy processes from Kuznetsov et al. (2011) to path functionals; in particular, first passage times, overshoots, undershoots, and the last maximum before the passage time. Such functionals have many applications, for instance, in finance (the pricing of exotic options in a Lévy model) and insurance (ruin time, debt at ruin, and related quantities for a Lévy insurance risk process). The technique works for any Lévy process whose running infimum and supremum evaluated at an independent exponential time can be sampled from. This includes classic examples such as stable processes, subclasses of spectrally one-sided Lévy processes, and large new families such as meromorphic Lévy processes. Finally, we present some examples. A particular aspect that is illustrated is that the Wiener-Hopf Monte Carlo simulation technique (provided that it applies) performs much better at approximating first passage times than a ‘plain’ Monte Carlo simulation technique based on sampling increments of the Lévy process.

This paper is concerned with the behaviour of a Lévy process when it crosses over a positive level, u, starting from 0, both as u becomes large and as u becomes small. Our main focus is on the time, τu, it takes the process to transit above the level, and in particular, on the stability of this passage time; thus, essentially, whether or not τu behaves linearly as u ↓ 0 or u → ∞. We also consider the conditional stability of τu when the process drifts to -∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process, which we analyse under a Cramér condition.

In the spirit of Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) we consider a Lévy insurance risk model with tax payments of a more general structure than in the aforementioned papers, which was also considered in Albrecher, Borst, Boxma, and Resing (2009). In terms of scale functions, we establish three fundamental identities of interest which have stimulated a large volume of actuarial research in recent years. That is to say, the two-sided exit problem, the net present value of tax paid until ruin, as well as a generalized version of the Gerber–Shiu function. The method we appeal to differs from Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) in that we appeal predominantly to excursion theory.

This paper is concerned with the finiteness and large-time behaviour of moments of the overshoot and undershoot of a high level, and of their moment generating functions (MGFs), for a Lévy process which drifts to -∞ almost surely. This provides information relevant to quantities associated with the ruin of an insurance risk process. Results of Klüppelberg, Kyprianou, and Maller (2004) and Doney and Kyprianou (2006) for asymptotic overshoot and undershoot distributions in the class of Lévy processes with convolution equivalent canonical measures are shown to have moment and MGF convergence extensions.

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), 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 of the nth 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 the 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 nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

In this short paper, we show how fluctuation identities for Lévy processes with no positive jumps yield the distribution of the present value of dividends paid until ruin in a Lévy insurance risk model with a dividend barrier.

In this short note we show how new fluctuation identities and their associated asymptotics, given in Vigon (2002), Klüppelberg et al. (2004) and Doney and Kyprianou (2006), provide the basis for establishing, in an elementary way, asymptotic overshoot and undershoot distribitions for a general class of Lévy insurance risk processes. The results bring the earlier conclusions of Asmussen and Klüppelberg (1996) for the Cramér-Lundberg process into greater generality.