Related papers: Levy Processes: Hitting time, overshoot and unders…

Let $a\in (0,\infty)$. For a spectrally negative L\'evy process $X$ with infinite variation paths the resolvent of the process killed on hitting the two-point set $V=\{-a,a\}$ is identified. When further $X$ has no diffusion component the…

We study the small-time asymptotics of sample paths of L\'evy processes and L\'evy-type processes. Namely, we investigate under which conditions the limit $$\limsup_{t \to 0} \frac{1}{f(t)} |X_t-X_0|$$ is finite resp.\ infinite with…

Let $\tau(x)$ be the epoch of first entry into the interval $(x,\infty)$, $x>0$, of the reflected process $Y$ of a L\'evy process $X$, and define the overshoot $Z(x) = Y(\tau(x))-x$ and undershoot $z(x) = x - Y(\tau(x)-)$ of $Y$ at the…

Let $\xi$ be a L\'{e}vy process and $I_\xi(t):=\int_{0}^te^{-\xi_s}\mathrm{d} s$, $t\geq 0,$ be the exponential functional of L\'{e}vy processes on deterministic horizon. Given that $\lim_{t\to \infty}\xi_t=-\infty$ we evaluate for general…

Consider the set of functions $f_{\theta}(x)=|\theta -x|$ on $\mathbb{R}$. Define a Markov process that starts with a point $x_0 \in \mathbb{R}$ and continues with $x_{k+1}=f_{\theta_{k+1}}(x_{k})$ with each $\theta _{k+1}$ picked from a…

Motivated by the recent results of Nualart and Xu \cite{Nualart} concerning limits laws for occupation times of one dimensional symmetric stable processes, this paper proves a decomposition for functionals of one dimensional symmetric…

We consider the problem of finding a stopping time that minimises the $L^1$-distance to $\theta$, the time at which a L\'evy process attains its ultimate supremum. This problem was studied in [12] for a Brownian motion with drift and a…

In this paper we study the exponential functionals of the processes $X$ with independent increments , namely $$I_t= \int _0^t\exp(-X_s)ds, _,\,\, t\geq 0,$$ and also $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ When $X$ is a…

We derive a criterium for the almost sure finiteness of perpetual integrals of \LL processes for a class of real functions including all continuous functions and for general one-dimensional L\'evy processes that drifts to plus infinity.…

In this paper, we will prove that the local time of a L\'evy process is of finite $p$-variation in the space variable in the classical sense, a.s. for any $p>2$, $t\geq 0$, if the L\'evy measure satisfies $\int_{R\setminus…

Let $X$ be a squared Bessel process. Following a Feynman-Kac approach, the Laplace transforms of joint laws of $(U, \int_0^{R_y}X_s^p\,ds)$ are studied where $R_y$ is the first hitting time of $y$ by $X$ and $U$ is a random variable…

The {\em drawdown} process $Y$ of a completely asymmetric L\'{e}vy process $X$ is equal to $X$ reflected at its running supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the…

Let $X$ be a symmetric, pure jump, unimodal Levy process in $\mathbb{R}$ with an infinite Levy measure. We prove that for any fixed $t > 0$ the survival probability $P^x(\tau_{(-a,a)} > t)$ is nondecreasing on $(-a,0]$, nonincreasing on…

We consider some special classes of L\'evy processes with no gaussian component whose L\'evy measure is of the type $\pi(dx)=e^{\gamma x}\nu(e^x-1) dx$, where $\nu$ is the density of the stable L\'evy measure and $\gamma$ is a positive…

Let $Y$ be a symmetric Borel right process with locally compact state space $T\subseteq R^{1}$ and potential densities $u(x,y)$ with respect to some $\sigma$-finite measure on $T$. Let $g$ and $f$ be finite excessive functions for $ Y$. Set…

We wish to characterise when a L\'{e}vy process $X_t$ crosses boundaries like $t^\kappa$, $\kappa>0$, in a one or two-sided sense, for small times $t$; thus, we enquire when $\limsup_{t\downarrow 0}|X_t|/t^{\kappa}$, $\limsup_{t\downarrow…

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…

We present an explicit solution to the Skorokhod embedding problem for spectrally negative L\'evy processes. Given a process $X$ and a target measure $\mu$ satisfying an explicit admissibility condition we define functions $\f_\pm$ such…

A refracted L\'evy process is a L\'evy 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…

We completely classify Fourier summation formulas of the form $$ \int_{\mathbb{R}} \widehat{\varphi}(t) d\mu(t)=\sum_{n=0}^{\infty} a(\lambda_n)\varphi(\lambda_n), $$ that hold for any test function $\varphi$, where $\widehat\varphi$ is the…