Related papers: Exponential functionals of L\'evy processes with j…

Let $X=(X_t)_{t\ge0}$ be a stable L\'{e}vy process of index $\alpha \in(1,2)$ with no negative jumps and let $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t>0$. We show that the density function $f_t$ of $S_t$ can be…

For $n$ equidistant observations of a L\'evy process at time distance $\Delta_n$ we consider the problem of testing hypotheses on the volatility, the jump measure and its Blumenthal-Getoor index in a non- or semiparametric manner.…

The dynamics of the eigenvalues (semimartingales) of a L\'{e}vy process $X$ with values in Hermitian matrices is described in terms of It\^{o} stochastic differential equations with jumps. This generalizes the well known Dyson-Brownian…

In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic L\'evy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise…

For a given L\'{e}vy process $X=(X_t)_{t\in\mathbb{R}_+}$ and for fixed $s\in \mathbb{R}_{+}\cup\{\infty\}$ and $t\in\mathbb{R}_+$ we analyse the {\it future drawdown extremes} that are defined as follows: \begin{eqnarray*} \overline…

For a L\'evy process on the real line, we provide complete criteria for the finiteness of exponential moments of the first passage time into the interval $(r,\infty)$, the sojourn time in the interval $(-\infty,r]$, and the last exit time…

We consider a stochastic volatility model with L\'evy jumps for a log-return process $Z=(Z_{t})_{t\geq 0}$ of the form $Z=U+X$, where $U=(U_{t})_{t\geq 0}$ is a classical stochastic volatility process and $X=(X_{t})_{t\geq 0}$ is an…

This article studies the dynamics of a nonlinear dissipative reaction-diffusion equation with well-separated stable states which is perturbed by infinite-dimensional multiplicative L\'evy noise with a regularly varying component at…

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…

We consider the linear growth-fragmentation equation arising in the modelling of cell division or polymerisation processes. For constant coefficients, we prove that the dynamics converges to the steady state with an exponential rate. The…

Motivated by the pricing of lookback options in exponential L\'evy models, we study the difference between the continuous and discrete supremum of L\'evy processes. In particular, we extend the results of Broadie et al. (1999) to…

In a high-frequency context, we investigate the efficient estimation of scaling and jump activity parameters for a stochastic differential equation driven by a L{\'e}vy process with both diffusion component and pure-jump component. We first…

Let ${\mathcal A}$ be the ${\mathcal L}^q-$functional of a stable L\'evy process starting from one and killed when crossing zero. We observe that ${\mathcal A}$ can be represented as the independent quotient of two infinite products of…

We study the work fluctuations of a particle subjected to a deterministic drag force plus a random forcing whose statistics is of the L\'evy type. In the stationary regime, the probability density of the work is found to have ``fat''…

Let $\{X_{1}(t)\}_{0\leq t\leq1}$ and $\{X_{2}(t)\}_{0\leq t\leq1}$ be two independent continuous centered Gaussian processes with covariance functions$R_{1}$ and $R_{2}$. This paper shows that if the covariance functions are of finite…

L\'evy walks (LWs) are spatiotemporally coupled random-walk processes describing superdiffusive heat conduction in solids, propagation of light in disordered optical materials, motion of molecular motors in living cells, or motion of…

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…

Gamma distributions, which contain the exponential as a special case, have a distinguished place in the representation of near-Poisson randomness for statistical processes; typically, they represent distributions of spacings between events…

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 provide exact large-time equivalents of the density and upper tail distributions of the exponential functional of a subordinator in terms of its Laplace exponents. This improves previous results on the logarithmic asymptotic behaviour of…