Related papers: Ray-Knight Theorems for Spectrally Negative L\'evy…

We consider the passage time problem for L\'evy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean case),…

Local time of a stochastic process quantifies the amount of time that sample trajectories $x(\tau)$ spend in the vicinity of an arbitrary point $x$. For a generic Hamiltonian, we employ the phase-space path-integral representation of random…

We present a simple stochastic integral representation for the local times of the height process of a spectrally positive Levy process stopped at a hitting time. From the representation we derive a strong stochastic equation for the local…

For a recurrent linear diffusion on $\R_+$ we study the asymptotics of the distribution of its local time at 0 as the time parameter tends to infinity. Under the assumption that the L\'evy measure of the inverse local time is subexponential…

In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential…

Several aspects of the laws of first hitting times of points are investigated for one-dimensional symmetric stable L\'evy processes. It\^o's excursion theory plays a key role in this study.

In this paper, we introduce the concept of \emph{Poissonian occupation times} below level $0$ of spectrally negative L\'evy processes. In this case, occupation time is accumulated only when the process is observed to be negative at arrival…

L\'{e}vy walk is a popular and more `physical' model to describe the phenomena of superdiffusion, because of its finite velocity. The movements of particles are under the influences of external potentials almost at anytime and anywhere. In…

In this paper we study the draw-down related Parisian ruin problem for spectrally negative L\'{e}vy risk processes. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit time via excursion theory. We also…

Long-time limit of one-dimensional L\'{e}vy processes weighted and normalized with respect to the exponential functional of two-point local times are studied. The limit processes may vary according to the choice of random clocks.

For a continuous-time random walk $X=\{X_t,t\ge 0\}$ (in general non-Markov), we study the asymptotic behavior, as $t\rightarrow \infty$, of the normalized additive functional $c_t\int_0^{t} f(X_s)ds$, $t\ge 0$. Similarly to the Markov…

For spectrally negative L\'evy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find…

For a stochastic process $(X_t)_{t\geq 0}$ we establish conditions under which the inverse first-passage time problem has a solution for any random variable $\xi >0$. For Markov processes we give additional conditions under which the…

We construct the analogue of the local time -- at a fixed point $x$ -- for Markov processes indexed by Levy trees. We start by proving that Markov processes indexed by Levy trees satisfy a special Markov property which can be thought as a…

In this paper we study a spectrally negative L\'{e}vy process that is reflected at its draw-down level whenever a draw-down time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we…

A step reinforced random walk is a discrete time process with memory such that at each time step, with fixed probability $p \in (0,1)$, it repeats a previously performed step chosen uniformly at random while with complementary probability…

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…

Distributional identities for a L\'evy process $X_t$, its quadratic variation process $V_t$ and its maximal jump processes, are derived, and used to make "small time" (as $t\downarrow0$) asymptotic comparisons between them. The…

This paper considers magnitude, asymptotics and duration of drawdowns for some L\'{e}vy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative L\'{e}vy processes using an approximation…

The subject of this paper is to prove a functional weak invariance principle for the local time of a process generated by a Gibbs-Markov map. More precisely, let $\left(X,\mathcal{B},m,T,\alpha\right)$ is a mixing, probability preserving…