275 research outputs found
Some singular sample path properties of a multiparameter fractional Brownian motion
We prove a Chung-type law of the iterated logarithm for a multiparameter
extension of the fractional Brownian motion which is not increment stationary.
This multiparameter fractional Brownian motion behaves very differently at the
origin and away from the axes, which also appears in the Hausdorff dimension of
its range and in the measure of its pointwise H\"older exponents. A functional
version of this Chung-type law is also provided.Comment: 21 pages. To appear in J. Theoret. Proba
Infinite Dimensional Pathwise Volterra Processes Driven by Gaussian Noise -- Probabilistic Properties and Applications
We investigate the probabilistic and analytic properties of Volterra
processes constructed as pathwise integrals of deterministic kernels with
respect to the H\"older continuous trajectories of Hilbert-valued Gaussian
processes. To this end, we extend the Volterra sewing lemma from
\cite{HarangTindel} to the two dimensional case, in order to construct two
dimensional operator-valued Volterra integrals of Young type. We prove that the
covariance operator associated to infinite dimensional Volterra processes can
be represented by such a two dimensional integral, which extends the current
notion of representation for such covariance operators. We then discuss a
series of applications of these results, including the construction of a rough
path associated to a Volterra process driven by Gaussian noise with possibly
irregular covariance structures, as well as a description of the irregular
covariance structure arising from Gaussian processes time-shifted along
irregular trajectories. Furthermore, we consider an infinite dimensional
fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be
seen as an extension of the volatility model proposed by Rosenbaum et al. in
\cite{ElEuchRosenbaum}.Comment: 38 page
The Sequential Empirical Process of a Random Walk in Random Scenery
A random walk in random scenery is given by
for a random walk and iid random
variables . In this paper, we will show the weak
convergence of the sequential empirical process, i.e. the centered and rescaled
empirical distribution function. The limit process shows a new type of
behavior, combining properties of the limit in the independent case (roughness
of the paths) and in the long range dependent case (self-similarity)
Skorohod and Stratonovich integration in the plane
This article gives an account on various aspects of stochastic calculus in
the plane. Specifically, our aim is 3-fold: (i) Derive a pathwise change of
variable formula for a path indexed by a square, satisfying some H\"older
regularity conditions with a H\"older exponent greater than 1/3. (ii) Get some
Skorohod change of variable formulas for a large class of Gaussian processes
defined on the suqare. (iii) Compare the bidimensional integrals obtained with
those two methods, computing explicit correction terms whenever possible. As a
byproduct, we also give explicit forms of corrections in the respective change
of variable formulas
Functional Limit Theorems for Volterra Processes and Applications to Homogenization
We prove an enhanced limit theorem for additive functionals of a
multi-dimensional Volterra process in the rough path
topology. As an application, we establish weak convergence as of the solution of the random ordinary differential equation (ODE)
and show that its limit solves a
rough differential equation driven by a Gaussian field with a drift coming from
the L\'evy area correction of the limiting rough driver. Furthermore, we prove
that the stochastic flows of the random ODE converge to those of the Kunita
type It\^o SDE , where is a semi-martingale with
spatial parameters.Comment: 31 page
Multi-dimensional parameter estimation of heavy-tailed moving averages
In this paper we present a parametric estimation method for certain
multi-parameter heavy-tailed L\'evy-driven moving averages. The theory relies
on recent multivariate central limit theorems obtained in [3] via Malliavin
calculus on Poisson spaces. Our minimal contrast approach is related to the
papers [14, 15], which propose to use the marginal empirical characteristic
function to estimate the one-dimensional parameter of the kernel function and
the stability index of the driving L\'evy motion. We extend their work to allow
for a multi-parametric framework that in particular includes the important
examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck
process, certain CARMA(2, 1) models and Ornstein-Uhlenbeck processes with a
periodic component among other models. We present both the consistency and the
associated central limit theorem of the minimal contrast estimator.
Furthermore, we demonstrate numerical analysis to uncover the finite sample
performance of our method
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