Right inverses of Lévy processes and stationary stopped local times (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;
Occupation Times of Refracted Lévy Processes
Journal of Theoretical Probability, 2013
A refracted Lévy process is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More precisely, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation dU t = −δ1 {Ut>b} dt + dX t , where X = (X t , t ≥ 0) is a Lévy process with law P and b, δ ∈ R such that the resulting process U may visit the half line (b, ∞) with positive probability. In this paper, we consider the case that X is spectrally negative and establish a number of identities for the following functionals ∞ 0
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Forum Mathematicum, 2020
In this paper, our first goal is to rigorously define a Lévy process pinned at random time. Our second task is to establish the Markov property with respect to its completed natural filtration and thus with respect to the usual augmentation of the latter. The resulting conclusion is the right-continuity of completed natural filtration. Certain examples of such process are considered.
Summary. The Levy system for a Markov process X provides a convenient descrip- tion of the distribution of the totally inaccessible jumps of the process. We examine the effect of time change (by the inverse of a not necessarily strictly increasing CAF A )o n the Lsystem, in a general context. They key to our time-change theorem is a study of the "irregular" exits from the fine support of A that occur at totally inaccessible times. This permits the construction of a partial predictable exit system (` a la Maisonneuve). The second part of the paper is devoted to some implications of the preceding in a (weak, moderate Markov) duality setting. Fixing an excessive measure m (to serve as duality measure) we obtain formulas relating the "killing" and "jump" measures for the time-changed process to the analogous objects for the original process. These formulas extend, to a very general context, recent work of Chen, Fukushima, and Ying. The key to our developmen...
Illinois Journal of Mathematics
It is well known that a class of subordinators can be represented using the local time of Brownian motions. An extension of such a representation is given for a class of Lévy processes which are not necessarily of bounded variation. This class can be characterized by the complete monotonicity of the Lévy measures. The asymptotic behavior of such processes is also discussed and the results are applied to the generalized arc-sine law, an occupation time problem on the positive side for one-dimensional diffusion processes.
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.
Asymptotic behaviour of first passage time distributions for Lévy processes
Probability Theory and Related Fields, 2013
In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt . In the particular case of a subordinator in the domain of attraction of a stable distribution our results are the analogue of the results obtained by the authors in [6] for non-monotone Lévy processes. For subordinators an approach different to that in [6] is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential. ‡ Research funded by the CONACYT Project Teoría y aplicaciones de procesos de Lévy where b denotes the drift and Π the Lévy measure of X. We will write ψ * for the exponent of {Xt − bt, t ≥ 0}, so that ψ * (λ) := ψ(λ) − bλ, λ ≥ 0.
Fluctuation theory and exit systems for positive self-similar Markov processes
Annals of Probability, 2012
For a positive self-similar Markov process, X, we construct a local time for the random set, , of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H ) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set and the process X sampled on the local time scale. The process (R, H ) is described in terms of a ladder process linked to the Lévy process associated to X via Lamperti's transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R, H ) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying Lévy process oscillates.
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.