Related papers: Generalized refracted L\'evy process and its appli…

We demonstrate the existence of a "L\'evy system" for the excursions of a one-dimensional diffusion process above its past-minimum process. As applications we provide a direct proof of D. Williams' decomposition (in both a global and a…

In this paper, we derive identities for the upward and downward exit problems and resolvents for a process whose motion changes between two L\'evy processes if it is above (or below) a barrier $b$ and coincides with a Poissonian arrival…

We develop a method that relates the truncated cumulant-function of the fourth order with the L\'evian cumulant-function. This gives us explicit formulas for the L\'evy-parameters, which allow a real-time analysis of the state of a…

In this paper we study a spectrally negative L\'evy process which is refracted at its running maximum and at the same time reflected from below at a certain level. Such a process can for instance be used to model an insurance surplus…

This paper solves exit problems for spectrally negative Markov additive processes and their reflections. A so-called scale matrix, which is a generalization of the scale function of a spectrally negative \levy process, plays a central role…

We consider a process $Z$ on the real line composed from a L\'evy process and its exponentially tilted version killed with arbitrary rates and give an expression for the joint law of $Z$ seen from its supremum, the supremum $\overline Z$…

In the setting of a L\'evy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold $r$. First, we give the joint…

In this paper we consider the argmin process of random walks and L\'evy processes. We prove that they enjoy the Markov property, and provide their transition kernels in some special cases.

We develop an excursion theory that describes the evolution of a Markov process indexed by a Levy tree away from a regular and instantaneous point $x$ of the state space. The theory builds upon a notion of local time at $x$ that was…

The L\'evy walk is a non-Brownian random walk model that has been found to describe anomalous dynamic phenomena in diverse fields ranging from biology over quantum physics to ecology. Recurrently occurring problems are to examine whether…

We consider a continuous-time random walk which is defined as an interpolation of a random walk on a point process on the real line. The distances between neighboring points of the point process are i.i.d. random variables in the normal…

For a L\'evy process $\xi=(\xi_t)_{t\geq0}$ drifting to $-\infty$, we define the so-called exponential functional as follows \[{\rm{I}}_{\xi}=\int_0^{\infty}e^{\xi_t} dt.\] Under mild conditions on $\xi$, we show that the following…

We provide, in a general setting, explicit solutions for optimal stopping problems that involve a diffusion process and its running maximum. Besides, a new feature includes absorbing boundaries that vary with the value of the running…

We consider Kallenberg's hypothesis on the characteristic function of a L\'{e}vy process and show that it allows the construction of weakly continuous bridges of the L\'{e}vy process conditioned to stay positive. We therefore provide a…

The first-exit time process of an inverse Gaussian L\'evy process is considered. The one-dimensional distribution functions of the process are obtained. They are not infinitely divisible and the tail probabilities decay exponentially. These…

In this paper, we study the law of the local time processes $(L_T^x(X),x\in \mathbb{R})$ associated to a spectrally negative L\'evy process $X$, in the cases $T=\tau_a^+$, the first passage time of $X$ above $a>0$ and $T=\tau(c)$, the first…

Various recent results on quantum L\'evy processes are presented. The first part provides an introduction to the theory of L\'evy processes on involutive bialgebras. The notion of independence used for these processes is tensor…

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of L\'evy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker,…

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…

This paper considers a L\'evy-driven queue (i.e., a L\'evy process reflected at 0), and focuses on the distribution of $M(t)$, that is, the minimal value attained in an interval of length $t$ (where it is assumed that the queue is in…