Related papers: Integral functionals for spectrally positive Levy …

In this paper we establish functional Erd\H{o}s-Renyi laws for L\'evy processes, i.e. limit theorems for sets of functions on [0,1] associated to their increments. First, we determine precise conditions under which, in a general framework,…

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…

We give several general theorems concerning positive definite solutions of Riemann-Hilbert problems on the real line. Furthermore, as an example, we apply our theory to the characteristic function of a class of L\'{e}vy processes and we…

We construct intrinsic on-and off-diagonal upper and lower estimates for the transition probability density of a L\'evy process in small time. By intrinsic we mean that such estimates reflect the structure of the characteristic exponent of…

Dilative stability generalizes the property of selfsimilarity for infinitely divisible stochastic processes by introducing an additional scaling in the convolution exponent. Inspired by results of Igl\'oi, we will show how dilatively stable…

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…

Ito's construction of Markovian solutions to stochastic equations driven by a L\'evy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the…

The improper stochastic integral $Z=\int_0^{\infty-}\exp(-X_{s-})dY_s$ is studied, where $\{(X_t, Y_t), t \geqslant 0 \}$ is a L\'evy process on $\mathbb R ^{1+d}$ with $\{X_t \}$ and $\{Y_t \}$ being $\mathbb R$-valued and $\mathbb R…

We establish integral tests and laws of the iterated logarithm at 0 and at $+\infty$, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation, time reversal arguments and on…

For a refracted L\'evy process driven by a spectrally negative L\'evy process, we use a different approach to derive expressions for its q-potential measures without killing. Unlike previous methods whose derivations depend on scale…

Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of…

A time-varying empirical spectral process indexed by classes of functions is defined for locally stationary time series. We derive weak convergence in a function space, and prove a maximal exponential inequality and a…

For spectrally positive L\'evy processes killed on exiting the half-line, existence of a quasi-stationary distribution is characterized by the exponential integrability of the exit time, the Laplace exponent and the non-negativity of the…

For a spectrally negative L\'evy process $X$, we study the following distribution: $$ \mathbb{E}_x \left[ \mathrm{e}^{- q \int_0^t \mathbf{1}_{(a,b)} (X_s) \mathrm{d}s } ; X_t \in \mathrm{d}y \right], $$ where $-\infty \leq a < b < \infty$,…

We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable L{\'e}vy processes with positive jumps. Assuming the branching mechanism is critical…

We study a first passage time of a L\'evy process over a positive constant level. In the spectrally negative case we give conditions for absolutely continuity of the distributions of the first passage times. The tail asymptotics of their…

We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric $\alpha $-stable L\'{e}vy processes, whose permanent residence in a finite interval on a line…

In this article we derive formulas for the probability $P(\sup_{t\leq T} X(t)>u)$ $T>0$ and $P(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally positive L\'evy process with infinite variation. The formulas are generalizations of the…

This paper provides a framework for investigations in fluctuation theory for L\'evy processes with matrix-exponential jumps. We present a matrix form of the components of the infinitely divisible factorization. Using this representation we…

We consider a new family of $\R^d$-valued L\'{e}vy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance \cite{cc}, \cite{ckp},…