Related papers: Small deviations of general L\'{e}vy processes
Statistical inference for stochastic processes based on high-frequency observations has been an active research area for more than a decade. One of the most well-known and widely studied problems is that of estimation of the quadratic…
As an analogue to the explicit formula in the stable case, the asymptotic behavior at the origin of the renormalized zero resolvent of one-dimensional L\'evy processes is studied under certain regular variation conditions on the…
Letting~$N=\left\{N(t), t\geq0\right\}$ be a standard Poisson process, Stroock~ \cite{Stroock-1981} constructed a family of continuous processes by $$\Theta_{\epsilon}(t)=\int_0^t\theta_{\epsilon}(r)dr, \ \ \ \ \ 0 \le t \le 1,$$ where…
This paper is concerned with asymptotic behavior (at zero and at infinity) of the favorite points of L\'evy processes. By exploring Molchan's idea for deriving lower tail probabilities of Gaussian processes with stationary increments, we…
We study the short-time asymptotical behavior of stochastic flows on \mathbb{R} in the \sup-norm. The results are stated in terms of a Gaussian process associated with the covariation of the flow. In case the Gaussian process has a…
Asymptotics deviation probabilities of the sum S n = X 1 + $\times$ $\times$ $\times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated, in particular when X 1 is not…
Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent, but numerically…
Particle approximations for certain nonlinear and nonlocal reaction-diffusion equations are studied using a system of Brownian motions with killing. The system is described by a collection of i.i.d. Brownian particles where each particle is…
We establish a large deviation principle for the solutions of a class of stochastic partial differential equations with non-Lipschitz continuous coefficients. As an application, the large deviation principle is derived for super-Brownian…
This article concerns the large deviations regime and the consequent solution of the Kramers problem for a two-time scale stochastic system driven by a common jump noise signal perturbed in small intensity $\varepsilon>0$ and with…
Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $\gamma$ be a non-negative constant. In this paper we study the asymptotics of $P\{\underset{t\in [0,\mathcal{T}/u^\gamma]}\sup X(t)>u\}$ as $u\to\infty$, with $\mathcal{T}$ an…
The goal of the paper is to analytically examine escape probabilities for dynamical systems driven by symmetric $\alpha$-stable L\'evy motions. Since escape probabilities are solutions of a type of integro-differential equations (i.e.,…
Let $A_\pm>0$, $\beta\in(0,1)$, and let $Z^{(\alpha)}$ be a strictly $\alpha$-stable L\'evy process with the jump measure $\nu(\mathrm{d} z)=(C_+\mathbb{I}_{(0,\infty)}(z)+ C_-\mathbb{I}_{(-\infty,0)}(z))|z|^{-1-\alpha}\,\mathrm{d} z$,…
The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable L\'evy processes, and fractional Brownian motion. For this reason it may be regarded as a basic building block for continuous time…
We obtain sample-path large deviations for a class of one-dimensional stochastic differential equations with bounded drifts and heavy-tailed L\'evy processes. These heavy-tailed L\'evy processes do not satisfy the exponential integrability…
We introduce a general theory on stationary approximations for locally stationary continuous-time processes. Based on the stationary approximation, we use $\theta$-weak dependence to establish laws of large numbers and central limit type…
In this paper we study the asymptotic theory for quadratic variation of a harmonizable fractional $\al$-stable process. We show a law of large numbers with a non-ergodic limit and obtain weak convergence towards a L\'evy-driven Rosenblatt…
L\'{e}vy processes with completely monotone jumps appear frequently in various applications of probability. For example, all popular stock price models based on L\'{e}vy processes (such as the Variance Gamma, CGMY/KoBoL and Normal Inverse…
We derive the exact asymptotics of $P(\sup_{u\leq t}X(u) > x)$ if $x$ and $t$ tend to infinity with $x/t$ constant, for a L\'{e}vy process $X$ that admits exponential moments. The proof is based on a renewal argument and a two-dimensional…
Using complex analysis techniques we obtain precise asymptotic approximations for the kernels corresponding to the symmetric $\alpha$-stable processes and their fractional derivatives. We apply our method to general L\'evy processes whose…