Page 556 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 556
DELING ROUGHNESS AND LONG-RANGE DEPENDENCE WITH FRACTIONAL
PROCESSES (MS-18)
Time-Changed Fractional Ornstein-Uhlenbeck Process
Giacomo Ascione, giacomo.ascione@unina.it
Università degli studi di Napoli Federico II, Italy
Coauthors: Enrica Pirozzi, Yuliya Mishura
In this talk I will focus on the description of some characteristics of a time-changed fractional
Ornstein-Uhlenbeck process, i. e. a process obtained by considering a fractional Ornstein-
Uhlenbeck process as introduced in [“Fractional Ornstein-Uhlenbeck Process" by Cheridito,
Kawaguchi and Maejima] and we substitute the time by the inverse of an independent driftless
subordinator. I will focus first on the existence and the asymptotics of the even-order moments.
Moreover, I will discuss some generalized Fokker-Planck equations that arise from the Gaussian
nature of the fractional Ornstein-Uhlenbeck process. Finally, I will focus on the stable case, for
which more explicit results can be achieved. Such things are the result of a joint work with
Yuliya Mishura from University of Kiev and Enrica Pirozzi from University of Naples and are
all contained in [“Time-changed fractional Ornstein- Uhlenbeck process", to appear].
Persistence probabilities of fractional processes
Frank Aurzada, aurzada@mathematik.tu-darmstadt.de
Technical University of Darmstadt, Germany
The talk will introduce the area of persistence probabilities. The questions studied in this area
are as follows: We are given a real-valued stochastic process, e.g. a fractional process such as
fractional Brownian motion. What is the probability that the process has a long excursion, e.g.
that is stays positive for a long time? What does the process look like if one conditions on
having a long excursion? These questions are classical for Brownian motion, random walks
and Lévy processes, with many applications in applied probability, such as queueing, finance,
insurance, etc. Contrary, for fractional processes, research on this type of problems has just
begun. We will survey the recent progess in this area.
Mild solutions of partial differential equations driven by general
stochastic measures
Iryna Bodnarchuk, ibodnarchuk@univ.kiev.ua
Taras Shevchenko National University of Kyiv, Ukraine
Coauthor: Vadym Radchenko
Let L0(Ω, F, P) be the set of all real-valued random variables defined on complete probability
space (Ω, F, P), X be an arbitrary set and B(X) be a σ-algebra of Borel subsets of X. Let µ be
a general stochastic measure, i.e., a σ-additive mapping µ : B(X) → L0(Ω, F, P).
We investigate Cauchy problems of a wave and heat equations driven by general stochastic
measures. The existence and uniqueness of the mild solutions are proved. Hölder regularity of
the paths in time and spatial variables is obtained. Asymptotic behavior of the mild solutions is
established.
554
PROCESSES (MS-18)
Time-Changed Fractional Ornstein-Uhlenbeck Process
Giacomo Ascione, giacomo.ascione@unina.it
Università degli studi di Napoli Federico II, Italy
Coauthors: Enrica Pirozzi, Yuliya Mishura
In this talk I will focus on the description of some characteristics of a time-changed fractional
Ornstein-Uhlenbeck process, i. e. a process obtained by considering a fractional Ornstein-
Uhlenbeck process as introduced in [“Fractional Ornstein-Uhlenbeck Process" by Cheridito,
Kawaguchi and Maejima] and we substitute the time by the inverse of an independent driftless
subordinator. I will focus first on the existence and the asymptotics of the even-order moments.
Moreover, I will discuss some generalized Fokker-Planck equations that arise from the Gaussian
nature of the fractional Ornstein-Uhlenbeck process. Finally, I will focus on the stable case, for
which more explicit results can be achieved. Such things are the result of a joint work with
Yuliya Mishura from University of Kiev and Enrica Pirozzi from University of Naples and are
all contained in [“Time-changed fractional Ornstein- Uhlenbeck process", to appear].
Persistence probabilities of fractional processes
Frank Aurzada, aurzada@mathematik.tu-darmstadt.de
Technical University of Darmstadt, Germany
The talk will introduce the area of persistence probabilities. The questions studied in this area
are as follows: We are given a real-valued stochastic process, e.g. a fractional process such as
fractional Brownian motion. What is the probability that the process has a long excursion, e.g.
that is stays positive for a long time? What does the process look like if one conditions on
having a long excursion? These questions are classical for Brownian motion, random walks
and Lévy processes, with many applications in applied probability, such as queueing, finance,
insurance, etc. Contrary, for fractional processes, research on this type of problems has just
begun. We will survey the recent progess in this area.
Mild solutions of partial differential equations driven by general
stochastic measures
Iryna Bodnarchuk, ibodnarchuk@univ.kiev.ua
Taras Shevchenko National University of Kyiv, Ukraine
Coauthor: Vadym Radchenko
Let L0(Ω, F, P) be the set of all real-valued random variables defined on complete probability
space (Ω, F, P), X be an arbitrary set and B(X) be a σ-algebra of Borel subsets of X. Let µ be
a general stochastic measure, i.e., a σ-additive mapping µ : B(X) → L0(Ω, F, P).
We investigate Cauchy problems of a wave and heat equations driven by general stochastic
measures. The existence and uniqueness of the mild solutions are proved. Hölder regularity of
the paths in time and spatial variables is obtained. Asymptotic behavior of the mild solutions is
established.
554
