On fluctuation theory for spectrally negative Lévy processes with Parisian reflection below, and applications

As is well known, all functionals of a Markov process may be expressed in terms of the generator operator, modulo some analytic work. In the case of spectrally negative Markov processes, however, it is conjectured that everything can be expressed in a more direct way using the $W$ scale function which intervenes in the two-sided first passage problem, modulo performing various integrals. This conjecture arises from work on Levy processes [6, 7, 12, 16, 28–30, 50] where the $W$ scale function has explicit Laplace transform, and is therefore easily computable; furthermore it was found in the papers above that a second scale function $Z$ introduced in [7] (this is an exponential transform (8) of $W$) greatly simplifies first passage laws, especially for reflected processes.

$Z$ is a harmonic function of the Lévy process (like $W$), corresponding to exterior boundary conditions $w(x)=e^{\theta x}$ (9) and is also a particular case of a “smooth Gerber–Shiu function” $\mathcal {S}_w$. The concept of the Gerber–Shiu function was introduced in [26]; we will use it however here in the more restricted sense of [15], who define this to be a “smooth” harmonic function of the process, which fits the exterior boundary condition $w(x)$ and simultaneously solves the problems (17), (18).

It has been conjectured that similar laws govern other classes of spectrally negativeprocesses, but it is quite difficult to find assumptions which allow proving this for general classes of Markov processes. However, we show below that in the particular case of spectrally negative Lévy processes with Parisian absorption and reflection from below [6, 16, 21], this conjecture holds true, once the appropriate $W$ and $Z$ are identified (this observation seems new).

This paper gathers a collection of first passage formulas for spectrally negative Parisian Lévy processes, expressed in terms of $W$, $Z$, and $\mathcal {S}_w$, which may serve as an “instruction kit” for computing quantities of interest in applications, for example in risk theory and mathematical finance. To illustrate the usefulness of our list, we construct a new index for the valuation of financial companies modeled by spectrally negative Lévy processes, based on a Dickson–Waters modifications of the de Finetti optimal expected discounted dividends objective. We offer as well an index for the valuation of conglomerates of financial companies.

An implicit question arising is to investigate analog results for other classes of spectrally negative Markovian processes.