Related papers: Almost sure asymptotics for the maximum local time…

We study a model of diffusion in a brownian potential. This model was firstly introduced by T. Brox (1986) as a continuous time analogue of random walk in random environment. We estimate the deviations of this process above or under its…

Let $X=\{X(t), t\geq 0\}$ be a Brownian motion or a spectrally negative stable process of index $1<\a<2$. Let $E=\{E(t),t\geq 0\}$ be the hitting time of a stable subordinator of index $0<\beta<1$ independent of $X$. We use a connection…

Let $X=(X_t)_{t\ge0}$ be a transient diffusion process in $(0,\infty)$ with the diffusion coefficient $\sigma>0$ and the scale function $L$ such that $X_t\rightarrow\infty$ as $t\rightarrow \infty$, let $I_t$ denote its running minimum for…

In this paper we consider non-local (in time) heat equations on time-increasing parabolic sets whose boundary is determined by a suitable curve. We provide a notion of solution for these equations and we study well-posedness under Dirichlet…

For a one-dimensional super-Brownian motion with density $X(t,x)$, we construct a random measure $L_t$ called the boundary local time which is supported on $\partial \{x:X(t,x) = 0\} =: BZ_t$, thus confirming a conjecture of Mueller, Mytnik…

In this paper, a class of statistics based on high frequency observations of oscillating and skew Brownian motion is considered. Their convergence rate towards the local time of the underlying process is obtained in form of a functional…

We consider two dependent Brownian motions with (possibly) different drift, and apply a result by le Gall on cone points of two dimensional Brownian motion to show that with probability one, there will not be a time that is a local maximum…

Based on an optimal rate wavelet series representation, we derive a local modulus of continuity result with a refined almost sure upper bound for fractional Brownian motion. \sloppy The obtained upper bound of the small fractional Brownian…

Let $W^H=\{W^H(t), t \in \rr\}$ be a fractional Brownian motion of Hurst index $H \in (0, 1)$ with values in $\rr$, and let $L = \{L_t, t \ge 0\}$ be the local time process at zero of a strictly stable L\'evy process $X=\{X_t, t \ge 0\}$ of…

We determine an asymptotic expression of the blow-up time t_coll for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point. We show that t_coll=t_{*}(eta-eta_c)^{-1/2} with…

Let $X=(X_t)_{t\geq 0}$ be a known process and $T$ an unknown random time independent of $X$. Our goal is to derive the distribution of $T$ based on an iid sample of $X_T$. Belomestny and Schoenmakers (2015) propose a solution based the…

In this note we investigate the behaviour of Brownian motion conditioned on a growth constraint of its local time which has been previously investigated by Berestycki and Benjamini. For a class of non-decreasing positive functions $f(t);…

In this paper, we consider a one-dimensional jump-type Cox-Ingersoll-Ross process driven by a Brownian motion and a subordinator, whose growth rate is an unknown parameter. Considering the process observed continuously or discretely at high…

Fractional Brownian motion is a non-Markovian Gaussian process $X_t$, indexed by the Hurst exponent $H$. It generalises standard Brownian motion (corresponding to $H=1/2$). We study the probability distribution of the maximum $m$ of the…

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold…

We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W\_\kappa$, with $ 0\textless{}\kappa\textless{}1$, and focus on the behavior of the local times $(\mathcal{L}(t,x),x)$ of $X$ before time $t\textgreater{}0$.In…

The problem is a log-asymptotics of the probability that the Integrated fractional Brownian motion of index 0<H<1 does not exceed a fixed level during long time. For the growing time interval (0,T) the hypothetical log-asymptotics is…

In this paper, we study the extinction time of logistic branching processes which are perturbed by an independent random environment driven by a Brownian motion. Our arguments use a Lamperti-type representation which is interesting on its…

We consider branching Brownian motion in which initially there is one particle at $x$, particles produce a random number of offspring with mean $m+1$ at the time of branching events, and each particle branches at rate $\beta = 1/2m$.…

In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a L\'evy process. More specifically, we investigate the asymptotic theory for the conditional mean…