Related papers: Tanaka formula for strictly stable processes

We study the existence and regularity of local times for general $d$-dimensional stochastic processes. We give a general condition for their existence and regularity properties. To emphasize the contribution of our results, we show that…

The problem of finding a necessary and sufficient condition for the continuity of the local times for a general Markov process is still open. Barlow and Hawkes have completely treated the case of the L\'{e}vy processes, and Marcus and Rosen…

In this brief note, we investigate some constructions of Lyapunov functions for stochastic discrete-time stabilizable dynamical systems, in other words, controlled Markov chains. The main question here is whether a Lyapunov function in some…

In this paper, we prove two limit laws for functionals of self-intersection symmetric alpha-stable processes with alpha\in(1,2). The results are obtained based on the method of moments, the sample configuration and the chaining argument…

We show that the derivative of the intersection and self-intersection local times of alpha-stable processes are exponentially integrable for certain parameter values. This includes the Brownian motion case. We also discuss related results…

We prove the existence of a local time, the continuity of the local time about $t$, and the regular property for $a.e.$ $x\in R$ of a Ornstein-Uhlenbeck type $\{X_t,\ t\in R^+\}$ driven by a general L\'{e}vy process, under mild regularity…

Suppose that $d\geq1$ and $\alpha\in (1, 2)$. Let $Y$ be a rotationally symmetric $\alpha$-stable process on $\R^d$ and $b$ a $\R^d$-valued measurable function on $\R^d$ belonging to a certain Kato class of $Y$. We show that $\rd X^b_t=\rd…

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…

In this paper, we present new results on finite- and fixed-time convergence for dynamical systems using LaSalle-like invariance principles. In particular, we provide first and second-order non-smooth Lyapunov-like results for finite- and…

Let $\eta=\{\eta(t);t\in [0,1]\}$ be a mean zero continuous Gaussian process with covariance $U=\{U(s,t),s,t\in [ 0,1]\},$ with $U(0,0)>0$. Let $\{\eta_{i};i=1,\ldots, k\}$ be independent copies of $\eta$ and set $ Y_{k}(t)=\sum_{i=1}^{k}…

We investigate the notion of finite time stability for tempered fractional systems (TFSs) with time delays and variable coefficients. Then, we examine some sufficient conditions that allow concluding the TFSs stability in a finite time…

We introduce two natural notions for the occupation measure of a function $V$ with finite variation. The first yields a signed measure, and the second a positive measure. By comparing two versions of the change-of-variables formula, we show…

Consider a symmetric $\alpha$-stable L\'evy process with $\alpha\in (1,2)$. We study shifted small ball probabilities for these processes in the uniform topology, when the shift function is an arbitrary continuous function which starts at…

Symmetric matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of…

We provide several characterizations of convergence to unstable equilibria in nonlinear systems. Our current contribution is three-fold. First we present simple algebraic conditions for establishing local convergence of non-trivial…

This paper considers the problem of finite-time stability for stochastic nonlinear systems. A new Lyapunov theorem of stochastic finite-time stability is proposed, and an important corollary is obtained. Some comparisons with the existing…

The literature on time series of functional data has focused on processes of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be…

Local time of a stochastic process quantifies the amount of time that sample trajectories $x(\tau)$ spend in the vicinity of an arbitrary point $x$. For a generic Hamiltonian, we employ the phase-space path-integral representation of random…

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 studies the inverse problem of determination the history for a stochastic diffusion process, by means of the value at the final time $T$. By establishing a new Carleman estimate, the conditional stability of the problem is…