Related papers: Continuity properties and the support of killed ex…

In this work we give a complete description to the asymptotic behaviors of exponential functionals of L\'evy processes and divide them into five different types according to their convergence rates. Not only their exact convergence speeds…

For a bivariate L\'evy process $(\xi_t,\eta_t)_{t\geq 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as V_t:=e^{\xi_t}(z+\int_0^t e^{-\xi_{s-}}d\eta_s), t\ge0, where $z\in\mathbb{R}.$ We define necessary and sufficient…

The $\alpha$-stable distributions introduced by L\'evy play an important role in probabilistic theoretical studies and their various applications, e.g., in statistical physics, life sciences, and economics. In the present paper we study…

Let $\xi_0,\xi_1,\ldots$ be independent identically distributed complex- valued random variables such that $\mathbb{E}\log(1+|\xi _0|)<\infty$. We consider random analytic functions of the form…

For a bivariate \Levy process $(\xi_t,\eta_t)_{t\geq 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as \[V_t:=e^{\xi_t}(z+\int_0^t e^{-\xi_{s-}}\ud \eta_s), t\ge0,\]where $z\in\mathbb{R}.$ We present conditions on the…

We demonstrate that two Ornstein--Uhlenbeck processes, that is, solutions to certain stochastic differential equations that are driven by a L\'evy process L have equivalent laws as long as the eigenvalues of the covariance operator…

For stochastic partial differential equations driven by L\'evy noise, understanding when changes in the drift operator preserve the law of the solution is fundamental to filtering, control, and simulation. We extend law-equivalence results…

In this paper we first provide several conditional limit theorems for L\'evy processes with negative drift and regularly varying tail. Then we apply them to study the asymptotic behavior of expectations of some exponential functionals of…

In this article we study the so-called cut-off phenomenon in the total variation distance when $n\to \infty$ for the family of continuous-time stochastic processes indexed by $n\in \mathbb{N}$, \[ \left( \mathcal{Z}^{(n)}_t=…

Consider a stable L\'evy process $X=(X_t,t\geq 0)$ and let $T_x$, for $x>0$, denote the first passage time of $X$ above the level $x$. In this work, we give an alternative proof of the absolute continuity of the law of $T_x$ and we obtain a…

In this paper, we consider the problem of statistical inference for generalized Ornstein-Uhlenbeck processes of the type \[ X_{t} = e^{-\xi_{t}} \left( X_{0} + \int_{0}^{t} e^{\xi_{u-}} d u \right), \] where \(\xi_s\) is a L{\'e}vy process.…

In this article, we first review the connection between L\'evy processes and infinitely divisible random variables, and the classification of infinitely divisible distributions. Using this connection and the L\'evy-Khinchine representation…

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 investigate the distributional properties of two generalized Ornstein-Uhlenbeck (OU) processes whose stationary distributions are the gamma law and the bilateral gamma law, respectively. The said distributions turn out to be related to…

In this paper we develop a framework for estimating Probability of Default (PD) based on stochastic models governing an appropriate asset value processes. In particular, we build upon a L\'evy-driven Ornstein-Uhlenbeck process and consider…

Let $u(s,t)$ be a continuous potential density of a symmetric L\'evy process or diffusion with state space $T$ killed at $T_{0}$, the first hitting time of $0$, or at $\lambda \wedge T_{0}$, where $\lambda$ is an independent exponential…

We consider infinitely divisible distributions with symmetric L\'evy measure and study the absolute continuity of them with respect to the Lebesgue measure. We prove that if $\eta(r)=\int_{|x|\le r} x^2 \nu(dx)$ where $\nu$ is the L\'evy…

In this paper, we study the asymptotic behavior of a supercritical $(\xi,\psi)$-superprocess $(X_t)_{t\geq 0}$ whose underlying spatial motion $\xi$ is an Ornstein-Uhlenbeck process on $\mathbb R^d$ with generator $L =…

We show some Chung-type $\liminf$ law of the iterated logarithm results at zero for a class of (pure-jump) Feller or L\'evy-type processes. This class includes all L\'evy processes. The norming function is given in terms of the symbol of…

Let $E$ be a space of observables in a sequence of trials $\xi_n$ and define $m_n$ to be the empirical distributions of the outcomes. We discuss the almost sure convergence of the sequence $m_n$ in terms of the $\psi$-weak topology of…