On Scale Functions for Spectrally Negative Lévy Processes with Phase-type Jumps (original) (raw)
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The Theory of Scale Functions for Spectrally Negative Lévy Processes
Lecture Notes in Mathematics, 2012
The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Lévy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Lévy processes, in particular a reasonable understanding of the Lévy-Khintchine formula and its relationship to the Lévy-Itô decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Lévy processes;
Special, conjugate and complete scale functions for spectrally negative Lévy processes
Electronic Journal of Probability, 2008
Following from recent developments in Hubalek and Kyprianou [30] the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative Lévy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative Lévy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed.
Smoothness of scale functions for spectrally negative Lévy processes
Probability Theory and Related Fields, 2011
Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation.
Theory of Probability and Mathematical Statistics
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 [AKP04, Pis05, APP07, Iva11, IP12, Iva13, AIZ16, APY16], 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 [AKP04] (this is an exponential transform (8) of W) greatly simplifies first passage laws, especially for reflected processes. Z is an harmonic function of the Lévy process (like W), corresponding to exterior boundary conditions w(x) = e θx (9), and is also a particular case of a "smooth Gerber-Shiu function" Sw. The concept of Gerber-Shiu function was introduced in [GS98]; we will use it however here in the more restricted sense of [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 (17), (18). It has been conjectured that similar laws govern other classes of spectrally negative 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 spectrally negative Lévy processes with Parisian absorption and reflection from below [AIZ16, BPPR16, APY16], 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 Sw, 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.
ESAIM: Probability and Statistics
In the last years there appeared a ``kit" of first passage identities for spectrally negative Levy processes, which are expressed in terms of two scale functions W and Z. Similar formulas are valid for: a) refracted processes (Kyprianou, Loeffen, Pardo, Perez, Renaud, Yamazaki: 2010, 2014, 2015 b) Markov additive processes: Ivanovs and Palmowski 2012, c) Levy processes with Poissonian Parisian absorbtion or/and reflection: Avram, Perez, Yamazaki, Zhou 2017, 2018, d) processes with Omega killing: Li, Palmowski, Czarna, Kaszubowski 2018, e) Levy driven Langevin processes: Czarna, Perez, Rolski, Yamazaki 2017 f) spectrally negative Markov processes for which certain limits exist: Landriault, Li, Zhang, Avram, 2017, 2018. g) positive self-similar Markov processes with one-sided jumps Vidmar(2018). We collect below our favorite recipes from the Levy “W,Z kit”, drawing from various applications in mathematical finance, risk, queueing, and inventory/storage theory. Applications concer...
Fluctuations of Omega-killed spectrally negative Lévy processes
Stochastic Processes and their Applications
In this paper we solve the exit problems for (reflected) spectrally negative Lévy processes, which are exponentially killed with a killing intensity dependent on the present state of the process and analyze respective resolvents. All identities are given in terms of new generalizations of scale functions. For the particular cases ω(x) = q and ω(x) = q1 (a,b) (x), we obtain results for the classical exit problems and the Laplace transforms of the occupation times in a given interval, until first passage times, respectively. Our results can also be applied to find the bankruptcy probability in the so-called Omega model, where bankruptcy occurs at rate ω(x) when the Lévy surplus process is at level x < 0. Finally, we apply the these results to obtain some exit identities for a spectrally positive self-similar Markov processes. The main method throughout all the proofs relies on the classical fluctuation identities for Lévy processes, the Markov property and some basic properties of a Poisson process.
Markov chain approximations to scale functions of Lévy processes
Stochastic Processes and their Applications, 2015
We introduce a general algorithm for the computation of the scale functions of a spectrally negative Lévy process X, based on a natural weak approximation of X via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with coefficients given explicitly in terms of the Lévy triplet of X. It is easy to implement and fast to execute. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of X and its scale functions, not unlike the one-dimensional Itô diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.
An explicit Skorokhod embedding for spectrally negative Levy processes
We present an explicit solution to the Skorokhod embedding problem for spectrally negative L\'evy processes. Given a process XXX and a target measure mu\mumu satisfying an explicit admissibility condition we define functions fpm\f_\pmfpm such that the stopping time T=inft>0:Xtin−f−(Lt),f+(Lt)T = \inf\{t>0: X_t \in \{-\f_-(L_t), \f_+(L_t)\}\}T=inft>0:Xtin−f−(Lt),f+(Lt) induces XTsimmuX_T\sim \muXTsimmu. We also treat versions of TTT which take into account the sign of the excursion straddling time ttt. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (1992) and we compute explicitly their general quantities in our setup. Our method relies on some new explicit calculations relating scale functions and the It\^o excursion measure of XXX. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.
Spectrally negative Lévy processes with Parisian reflection below and classical reflection above
Stochastic Processes and their Applications, 2017
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.