Page 558 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 558
DELING ROUGHNESS AND LONG-RANGE DEPENDENCE WITH FRACTIONAL
PROCESSES (MS-18)
the existence of such a moment for a homogeneous Markov chain is a drift condition of the form
P V ≤ λV + bIC, λ < 1. We generalized this result to the time-inhomogeneous case and proved
that it is sufficient to have a similar drift condition with different λt at different time steps t. We
showed that homogeneous condition λ < 1 could be relaxed in the time inhomogeneous case.
The second result of the presentation is related to studying the simultaneous hitting time
for an atom α by two time-inhomogeneous Markov chains. We established conditions for the
existence of the exponential moment for the hitting time and found computable bounds using
the drift condition described above.
Exact spectral asymptotics of fractional processes and its applications
Marina Kleptsyna, Marina.Kleptsyna@univ-lemans.fr
University of Le Mans, France
Coauthors: Pavel Chigansky, Dmytro Marushkevych
Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance
operator. However, eigenproblems are notoriously hard to solve explicitly and closed form
solutions are known only in a limited number of cases. In this talk we set up a framework for
the spectral analysis of the fractional type covariance operators, corresponding to an important
family of processes, which includes the fractional Brownian motion, its noise, the fractional
Ornstein–Uhlenbeck process and the integrated fractional Brownian motion. We obtain accurate
asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a
key to several problems, whose solution is long known in the standard Brownian case, but was
missing in the more general fractional setting. This includes computation of the exact limits of
L2-small ball probabilities and asymptotic analysis of singularly perturbed integral equations,
arising in mathematical physics and applied probability.
Log-periodically disturbed fractional calculus
Svenja Lage, Svenja.Lage@hhu.de
Heinrich-Heine University Düsseldorf, Germany
It is well-known that stable distributions solve particular fractional diffusion equations. In this
talk, we develop a similar connection between semistable densities and diffusion equations
involving log-periodically disturbed fractional derivatives. Starting from this connection, we
discuss the properties of these operators, which allow us to model log-periodically disturbed
long-range dependencies. Furthermore, we solve corresponding diffusion equations and apply
our theory to real-world applications.
556
PROCESSES (MS-18)
the existence of such a moment for a homogeneous Markov chain is a drift condition of the form
P V ≤ λV + bIC, λ < 1. We generalized this result to the time-inhomogeneous case and proved
that it is sufficient to have a similar drift condition with different λt at different time steps t. We
showed that homogeneous condition λ < 1 could be relaxed in the time inhomogeneous case.
The second result of the presentation is related to studying the simultaneous hitting time
for an atom α by two time-inhomogeneous Markov chains. We established conditions for the
existence of the exponential moment for the hitting time and found computable bounds using
the drift condition described above.
Exact spectral asymptotics of fractional processes and its applications
Marina Kleptsyna, Marina.Kleptsyna@univ-lemans.fr
University of Le Mans, France
Coauthors: Pavel Chigansky, Dmytro Marushkevych
Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance
operator. However, eigenproblems are notoriously hard to solve explicitly and closed form
solutions are known only in a limited number of cases. In this talk we set up a framework for
the spectral analysis of the fractional type covariance operators, corresponding to an important
family of processes, which includes the fractional Brownian motion, its noise, the fractional
Ornstein–Uhlenbeck process and the integrated fractional Brownian motion. We obtain accurate
asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a
key to several problems, whose solution is long known in the standard Brownian case, but was
missing in the more general fractional setting. This includes computation of the exact limits of
L2-small ball probabilities and asymptotic analysis of singularly perturbed integral equations,
arising in mathematical physics and applied probability.
Log-periodically disturbed fractional calculus
Svenja Lage, Svenja.Lage@hhu.de
Heinrich-Heine University Düsseldorf, Germany
It is well-known that stable distributions solve particular fractional diffusion equations. In this
talk, we develop a similar connection between semistable densities and diffusion equations
involving log-periodically disturbed fractional derivatives. Starting from this connection, we
discuss the properties of these operators, which allow us to model log-periodically disturbed
long-range dependencies. Furthermore, we solve corresponding diffusion equations and apply
our theory to real-world applications.
556
