Related papers: Distributional properties of exponential functiona…

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…

Let $\tau(x)$ be the first time the reflected process $Y$ of a Levy processes $X$ crosses x>0. The main aim of the paper is to investigate the asymptotic dependence of the path functionals: $Y(t) = X(t) - \inf_{0\leq s\leq t}X(s)$,…

Probability distributions defined on the half space are known to be quite different from those in the full space. Here, a nonextensive entropic treatment is presented for the half space in an analytic and self-consistent way. In this…

We study the distribution of the positive sojourn time $$ A_t:= \int_0^t \mathbf 1\{ X_s>0 \}ds $$ of an arbitrary L\'evy process $X:= (X_t)_{t\geq 0}$. For an exponential random variable $E^{(q)}$ of rate $q>0$ independent of $X$ we show…

We propose a simple conjecture for the functional form of the asymptotic behavior of work distributions for driven overdamped Brownian motion of a particle in confining potentials. This conjecture is motivated by the fact that these…

Let us consider a solution of the time-inhomogeneous stochastic differential equation driven by a Brownian motion with drift coefficient $b(t,x)=\rho\,{\rm sgn}(x)|x|^\alpha/t^\beta$. This process can be viewed as a distorted Brownian…

In this paper, we study nonparametric estimation of the L\'{e}vy density for L\'{e}vy processes, with and without Brownian component. For this, we consider $n$ discrete time observations with step $\Delta$. The asymptotic framework is: $n$…

We prove asymptotic behaviour of transition density for a large class of spectrally one-sided L\'evy processes of unbounded variation satisfying mild condition imposed on the second derivative of the Laplace exponent, or equivalently, on…

The present paper is concerned with the integral of the absolute value of a Brownian motion with drift. By establishing an asymptotic expansion of the space Laplace transform, we obtain series representations for the probability density…

Extending It\^o's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-It\^o, applies to one dimensional semimartingales and convex functions.…

Let $f$ be the density function associated to a matrix-exponential distribution of parameters $(\alpha, T,s)$. By exponentially tilting $f$, we find a probabilistic interpretation which generalises the one associated to phase-type…

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables (not necessarily identically distributed), and $\eta$ be a counting random variable independent of this sequence. We obtain sufficient conditions on…

Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion, to which we associate the exponential additive functional $A_{t}=\int _{0}^{t}e^{2B_{s}}ds,\,t\ge 0$. Starting from a simple observation of generalized inverse…

Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83--109] prove an extension of It\^{o}'s formula for $F(X_t,t)$, where $F(x,t)$ has a locally square-integrable derivative in $x$ that satisfies a mild continuity condition in $t$ and…

Motivated by a common Mathematical Finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of an associated Brownian motion. We prove a conjecture…

We investigate the distribution properties of the fractional L\'evy motion. We consider separately the cases $0<H<1/2$ (short memory) and $1/2<H<1$ (long memory), where $H$ is the Hurst parameter, and present the asymptotic behaviour of the…

This article deals with IDT processes, i.e. processes which are infinitely divisible with respect to time. Given an IDT process $(X_{t},\,t\geq0)$, there exists a unique (in law) L\'evy process $(L_{t}; t\geq0)$ which has the same…

Stretched-exponential relaxation is a widely observed phenomenon found in ordered ferromagnets as well as glassy systems. One modeling approach connects this behavior to a droplet dynamics described by an effective Langevin equation for the…

We prove several necessary and sufficient conditions for the existence of (smooth) transition probability densities for L\'evy processes and isotropic L\'evy processes. Under some mild conditions on the characteristic exponent we calculate…

We consider a diffusion process $X$ in a random potential $\V$ of the form $\V_x = \S_x -\delta x$ where $\delta$ is a positive drift and $\S$ is a strictly stable process of index $\alpha\in (1,2)$ with positive jumps. Then the diffusion…