31 research outputs found
A set-indexed Ornstein-Uhlenbeck process
The purpose of this article is a set-indexed extension of the well-known
Ornstein-Uhlenbeck process. The first part is devoted to a stationary
definition of the random field and ends up with the proof of a complete
characterization by its -continuity, stationarity and set-indexed Markov
properties. This specific Markov transition system allows to define a general
\emph{set-indexed Ornstein-Uhlenbeck (SIOU) process} with any initial
probability measure. Finally, in the multiparameter case, the SIOU process is
proved to admit a natural integral representation.Comment: 13 page
The Multiparameter Fractional Brownian Motion
We define and study the multiparameter fractional Brownian motion. This
process is a generalization of both the classical fractional Brownian motion
and the multiparameter Brownian motion, when the condition of independence is
relaxed. Relations with the L\'evy fractional Brownian motion and with the
fractional Brownian sheet are discussed. Different notions of stationarity of
the increments for a multiparameter process are studied and applied to the
fractional property. Using self-similarity we present a characterization for
such processes. Finally, behavior of the multiparameter fractional Brownian
motion along increasing paths is analysed.Comment: 9 page
Local H\"older regularity for set-indexed processes
In this paper, we study the H\"older regularity of set-indexed stochastic
processes defined in the framework of Ivanoff-Merzbach. The first key result is
a Kolmogorov-like H\"older-continuity Theorem, whose novelty is illustrated on
an example which could not have been treated with anterior tools. Increments
for set-indexed processes are usually not simply written as , hence we
considered different notions of H\"older-continuity. Then, the localization of
these properties leads to various definitions of H\"older exponents, which we
compare to one another.
In the case of Gaussian processes, almost sure values are proved for these
exponents, uniformly along the sample paths. As an application, the local
regularity of the set-indexed fractional Brownian motion is proved to be equal
to the Hurst parameter uniformly, with probability one.Comment: 32 page
A Characterization of the Set-indexed Fractional Brownian Motion by Increasing Paths
We prove that a set-indexed process is a set-indexed fractional Brownian
motion if and only if its projections on all the increasing paths are
one-parameter time changed fractional Brownian motions. As an application, we
present an integral representation for such processes.Comment: 6 page
Stochastic 2-microlocal analysis
A lot is known about the H\"older regularity of stochastic processes, in
particular in the case of Gaussian processes. Recently, a finer analysis of the
local regularity of functions, termed 2-microlocal analysis, has been
introduced in a deterministic frame: through the computation of the so-called
2-microlocal frontier, it allows in particular to predict the evolution of
regularity under the action of (pseudo-) differential operators. In this work,
we develop a 2-microlocal analysis for the study of certain stochastic
processes. We show that moments of the increments allow, under fairly general
conditions, to obtain almost sure lower bounds for the 2-microlocal frontier.
In the case of Gaussian processes, more precise results may be obtained: the
incremental covariance yields the almost sure value of the 2-microlocal
frontier. As an application, we obtain new and refined regularity properties of
fractional Brownian motion, multifractional Brownian motion, stochastic
generalized Weierstrass functions, Wiener and stable integrals.Comment: 35 page
From almost sure local regularity to almost sure Hausdorff dimension for Gaussian fields
Fine regularity of stochastic processes is usually measured in a local way by
local H\"older exponents and in a global way by fractal dimensions. Following a
previous work of Adler, we connect these two concepts for multiparameter
Gaussian random fields. More precisely, we prove that almost surely the
Hausdorff dimensions of the range and the graph in any ball are
bounded from above using the local H\"older exponent at . We define the
deterministic local sub-exponent of Gaussian processes, which allows to obtain
an almost sure lower bound for these dimensions. Moreover, the Hausdorff
dimensions of the sample path on an open interval are controlled almost surely
by the minimum of the local exponents.
Then, we apply these generic results to the cases of the multiparameter
fractional Brownian motion, the multifractional Brownian motion whose
regularity function is irregular and the generalized Weierstrass function,
whose Hausdorff dimensions were unknown so far.Comment: 28 page
2-microlocal analysis of martingales and stochastic integrals
Recently, a new approach in the fine analysis of stochastic processes sample
paths has been developed to predict the evolution of the local regularity under
(pseudo-)differential operators. In this paper, we study the sample paths of
continuous martingales and stochastic integrals. We proved that the almost sure
2-microlocal frontier of a martingale can be obtained through the local
regularity of its quadratic variation. It allows to link the H\"older
regularity of a stochastic integral to the regularity of the integrand and
integrator processes. These results provide a methodology to predict the local
regularity of diffusions from the fine analysis of its coefficients. We
illustrate our work with examples of martingales with unusual complex
regularity behavior and square of Bessel processes.Comment: 40 pages, 3 figure