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# Theory of Probability and Mathematical Statistics

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

### Florin Avram, Xiaowen Zhou.

Abstract: As 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, 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 \cite{AKP} (this is an exponential transform \eqref{Z} of $W$) greatly simplifies \fp laws, especially for reflected processes. $Z$ is an harmonic function of the L\'evy process (like $W$), corresponding to exterior boundary conditions $w(x)=e^{\th x}$ \eqref{hext}, and is also a particular case of a "smooth \GS function" $\mS_w$. The concept of \GS function was introduced in \cite{gerber1998time}; we will use it however here in the more restricted sense of \cite{APP15}, who define this to be a "smooth" harmonic function of the process, which fits the exterior boundary condition $w(x)$ and solves simultaneously the problems \eqref{serui}, \eqref{rserui}. It has been conjectured that similar laws govern other classes of \sn processes, 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 \sn \lev processes with Parisian absorption and reflection from below \cite{AIZ,BPPR,APY}, 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 L\'evy processes, expressed in terms of $W,Z$ and $\mS_w$, which may serve as an "instruction kit" for computing quantities of interest in applications, for example in risk theory and mathematical finance.\iffalse An important point to notice is that the \fp laws reviewed concern equally absorbed, reflected, or doubly reflected processes. The benefit of presenting together all the possible boundary behaviors, as is done already in the papers cited above, is that the results suggest interesting relationships %between the different boundary behaviors -- see for example \eqref{proof}.\fi To illustrate the usefulness of our list, we construct a new index for the valuation of financial companies modeled by spectrally negative L\'evy 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 inancial companies. %Analogue results for spectrally negative Markov additive processes are listed in the web version of the paper; An implicit question arising is to investigate analog results for other classes of \sn Markovian processes.

Keywords: spectrally negative Levy process, scale functions, capital injections, dividend optimization, valuation problem, Parisian absorbtion and reflection

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