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As a generalization of scale functions of spectrally negative L\'evy processes, we define scale functions of general standard processes with no positive jumps. For this purpose, we utilize excursion measures. Using our new scale functions…
We provide a novel expression of the scale function for a L\'evy processes with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for…
For a spectrally negative L\'evy process, scale functions appear in the solution of two-sided exit problems, and in particular in relation with the Laplace transform of the first time it exits a closed interval. In this paper, we consider…
Scale functions play a central role in the fluctuation theory of spectrally negative L\'evy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in…
We obtain series expansions of the $q$-scale functions of arbitrary spectrally negative L\'evy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit $q$-scale functions.…
We introduce a general algorithm for the computation of the scale functions of a spectrally negative L\'evy process $X$, based on a natural weak approximation of $X$ via upwards skip-free continuous-time Markov chains with stationary…
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 Levy processes. Our review also includes the first extensive overview of how to work numerically…
A systematic exposition of scale functions is given for positive self-similar Markov processes (pssMp) with one-sided jumps. The scale functions express as convolution series of the usual scale functions associated with spectrally one-sided…
For a spectrally negative L\'evy process with Laplace transform $\psi$, the $q$-scale function is characterized as the function whose Laplace transform is $(\psi(\cdot)-q)^{-1}$. It has applications in fluctuation theory, for example, exit…
Scale functions play a central role in the fluctuation theory of spectrally negative L\'evy processes. For spectrally negative compound Poisson processes with positive drift, a new representation of the $q$-scale functions in terms of the…
We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Levy processes. From this we introduce a general method for generating new families of scale functions. Using this method we…
The scale function holds significant importance within the fluctuation theory of Levy processes, particularly in addressing exit problems. However, its definition is established through the Laplace transform, thereby lacking explicit…
For a refracted spectrally negative Levy process, we find some new and fantastic formulas for its q-potential measures without killing. Unlike previous results, which are written in terms of the known q-scale functions, our formulas are…
We study the scale function of the spectrally negative phase-type Levy process. Its scale function admits an analytical expression and so do a number of its fluctuation identities. Motivated by the fact that the class of phase-type…
A level-dependent L\'evy process solves the stochastic differential equation $dU(t) = dX(t)-{\phi}(U(t)) dt$, where $X$ is a spectrally negative L\'evy process. A special case is a multi-refracted L\'evy process with…
In this paper, we compute the Laplace transform of occupation times (of the negative half-line) of spectrally negative L\'evy processes. Our results are extensions of known results for standard Brownian motion and jump-diffusion processes.…
Following from recent developments by Hubalek and Kyprianou, the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative L\'evy processes which are completely explicit.…
A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free L\'evy chains, i.e. for right-continuous random walks embedded into continuous time as compound Poisson processes. This is done by…
Pure-jump L\'evy processes are popular classes of stochastic processes which have found many applications in finance, statistics or machine learning. In this paper, we propose a novel family of self-decomposable L\'evy processes where one…
Scaling properties of time series are usually studied in terms of the scaling laws of empirical moments, which are the time average estimates of moments of the dynamic variable. Nonlinearities in the scaling function of empirical moments…