Related papers: A Generalized Occupation Time Formula For Continuo…
Inspired by coarea formula in geometric measure theory, an occupation time formula for continuous semimartingales in $\mathbb{R}^{N}$ is proven. The occupation measure of a semimartingale, for $N\geq2$, is singular with respect to Lebesgue…
We obtain functional central limit theorems for both discrete time expressions of the form $1/\sqrt{N}\sum_{n=1}^{[Nt]}(F(X(q_1(n)),\ldots, X(q_{\ell}(n)))-\bar{F})$ and similar expressions in the continuous time where the sum is replaced…
In this paper we generalize a representation formula for the local time of a function of a semimartingale due to Coquet and Ouknine \cite{Ouknine} , our formula being a pointwise equality between two processes we show in addition that the…
The approximation of integral type functionals is studied for discrete observations of a continuous It\^o semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for $L^2$-Sobolev functions…
Our aim is to unify and extend the large deviation upper and lower bounds for the occupation times of a Markov process with $L_2$ semigroups under minimal conditions on the state space and the process trajectories; for example, no strong…
Let $X=\{X_n: n\in\mathbb{N}\}$ be a long memory linear process in which the coefficients are regularly varying and innovations are independent and identically distributed and belong to the domain of attraction of an $\alpha$-stable law…
Generalised Ito formulae are proved for time dependent functions of continuous real valued semi-martingales. The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded…
This paper is concerned with asymptotic behavior of a variety of functionals of increments of continuous semimartingales. Sampling times are assumed to follow a rather general discretization scheme. If an underlying semimartingale is…
The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional c\`adl\`ag process. Upper bounds on the error are obtained under weak assumptions, generalizing previous…
Given a reference filtration $\mathbb{F}$, we develop in this work a generic method for computing the semimartingale decomposition of $\mathbb{F}$-martingales in some specific enlargements of $\mathbb{F}$. This method is then applied to the…
This paper presents a set of results relating to the occupation time $\alpha(t)$ of a process $X(\cdot)$. The first set of results concerns exact characterizations of $\alpha(t)$ for $t\geq0$, e.g., in terms of its transform up to an…
Assume that $T$ is a conservative ergodic measure preserving transformation of the infinite measure space $(X,\mathcal{A},\mu)$.We study the asymptotic behaviour of occupation times of certain subsets of infinite measure. Specifically, we…
We study a notion of generalized H\"older continuity for functions on $\mathbb{R}^d$. We show that for any bounded function $f$ of bounded support and any $r>0$, the $r$-oscillation of $f$ defined as $osc_r f (x):= \sup_{B_r(x)} f -…
We obtain the fluctuations for the occupation time of one-dimensional symmetric exclusion processes with speed change, where the transition rates (conductances) are driven by a general function W. The approach does not require sharp bounds…
In the paper the rescaled occupation time fluctuation process of a certain empirical system is investigated. The system consists of particles evolving independently according to \alpha-stable motion in R^d, \alpha<d<2\alpha. The particles…
On a probability space $(\Omega,\mathcal{A},\mathbb{Q})$ we consider two filtrations $\mathbb{F}\subset \mathbb{G}$ and a $\mathbb{G}$ stopping time $\theta$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$…
We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with…
Functional limit theorems are presented for the rescaled occupation time fluctuations process of a critical finite variance branching particle system in $R^d$ with symmetric a-stable motion starting off from either a standard Poisson random…
We describe the classes of functions $f=(f(x), x\in R)$, for which processes $f(W_t)-Ef(W_t)$ and $f(W_t)/Ef(W_t)$ are martingales. We apply these results to give a martingale characterization of general solutions of the quadratic and the…
We prove a conditional local limit theorem for discrete-time fractional Brownian motions (dfBm) with Hurst parameter 3/4<H<1. Using results from infinite ergodic theory it is then shown that the properly scaled occupation time of dfBm…