Spectrally negative Lévy processes with Parisian reflection below and classical reflection above

Abstract

We consider a company that receives capital injections so as to avoid ruin. Differently from the classical bail-out settings, where the underlying process is restricted to stay at or above zero, we study the case bail-out can only be made at independent Poisson observation times. Namely, we study a version of the reflected process that is pushed up to zero only on Poisson arrival times at which the process is below zero. We also study the case with additional classical reflection above so as to model a company that pays dividends according to a barrier strategy. Focusing on the spectrally negative Lévy case, we compute, using the scale function, various fluctuation identities, including capital injections and dividends.

Generalized refracted Lévy process and its application to exit problem

“…. As one of our main theorems, we show the Laplace transform (1) for the process Z = U, our new refracted Lévy process, takes the same form as (1) where W (q) U will be defined in Theorem 6.2 in a more complicated form than (1). Note that W (q) U 's will be represented using only Laplace exponents and scale functions of X and Y .…”

“…x by (1) with n X being replaced by P X x . We can therefore construct a Feller process from n U together with the family of stopped processes P U 0 x…”

TheW,Zscale functions kit for first passage problems of spectrally negative Lévy processes, and applications to control problems

“…Somewhat surprisingly, it appeared recently that the recipes reviewed below apply equally to spectrally negative Lévy processes with (exponential) Parisian absorbtion or reflection below [LRZ14a,AIZ16,AI15,PY15,BPPR16,APY16], once the appropriate scale functions W, Z identified in [APY16,AZ17]. This mystery was explained in [LP16,LZ17], who showed that the W, Z recipes appropriately extended apply to the general class of Omega models, of which exponential Parisian models are a particular case.…”

Optimal Periodic Dividend Strategies for Spectrally Positive Lévy Risk Processes With Fixed Transaction Costs

“…Specifically, we will borrow the recent results from the Section 6 of Pérez and Yamazaki (2018). To fully understand the results, we recommend the work of Bertoin (1998), Kyprianou (2014), Loeffen, Renaud and Zhou (2014), Pantí (2017), Chaumont and Doney (2005), Pardo, Pérez and Rivero (2015), Avram, Pérez and Yamazaki (2018) and Pérez and Yamazaki (2018) (in the order), together with the references therein.…”