Page 557 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 557
DELING ROUGHNESS AND LONG-RANGE DEPENDENCE WITH FRACTIONAL
PROCESSES (MS-18)
Recent developments in stochastic analysis of Rosenblatt processes
Petr Cˇ oupek, coupek@karlin.mff.cuni.cz
Charles University, Faculty of Mathematics and Physics, Czech Republic
The talk will be devoted to some recent developments in stochastic analysis of Rosenblatt and
related fractional processes.
Rosenblatt processes arise naturally as limits of suitably normalized sums of long-range
dependent random variables in non-central limit theorems. They share many properties with
regular fractional Brownian motions; in particular, they have stationary increments and they ex-
hibit self-similarity and long-range dependence. However, unlike fractional Brownian motions,
Rosenblatt process are not Gaussian which makes their analysis somewhat intriguing and which
is also the reason why they have received considerable attention in recent years.
In the talk, some recent results regarding Rosenblatt processes will be given and an approach
to stochastic integration for Rosenblatt processes that is based on the Malliavin calculus will be
discussed. In particular, an Itô-type stochastic chain rule for processes with a second-order
fractional stochastic differential will be presented.
Rough volatility: SDE driven by Hölder continuous noise and unbounded
drift
Giulia Di Nunno, giulian@math.uio.no
University of Oslo, Norway
Coauthors: Yuliya Mishura, Anton Yurchenko-Tytarenko
Having in mind possible stochastic volatility models in finance, we consider an SDE driven by
a general Hölder continuous noise. The drift b is exploding with a control from below:
b(t, y) > CT (y − ϕ(t))−γ (γ > 0),
where ϕ is a continuous function and CT , γ are constants. We study the solution of such SDE
and its properties. In particular, we prove that it has a unique solution which is bound preserv-
ing.
Furthermore, modifying the controls on the drift, we obtain an SDE sandwiched between
two given bounds ϕ and ψ, with ψ(x) > ϕ(x). This sandwich solution turns out to be the most
useful for applications.
Among the properties presented, we show that the solution admits all moments. This re-
markable result paves the way for efficient numerical methods.
Exponential moments of hitting times for time-inhomogeneous atomic
Markov chains
Vitaliy Golomoziy, vitaliy.golomoziy@gmail.com
Taras Shevchenko National University of Kyiv, Ukraine
The result we present is devoted to studying exponential moments of hitting times for time-
inhomogeneous Markov chains. It is well-known that a necessary and sufficient condition for
555
PROCESSES (MS-18)
Recent developments in stochastic analysis of Rosenblatt processes
Petr Cˇ oupek, coupek@karlin.mff.cuni.cz
Charles University, Faculty of Mathematics and Physics, Czech Republic
The talk will be devoted to some recent developments in stochastic analysis of Rosenblatt and
related fractional processes.
Rosenblatt processes arise naturally as limits of suitably normalized sums of long-range
dependent random variables in non-central limit theorems. They share many properties with
regular fractional Brownian motions; in particular, they have stationary increments and they ex-
hibit self-similarity and long-range dependence. However, unlike fractional Brownian motions,
Rosenblatt process are not Gaussian which makes their analysis somewhat intriguing and which
is also the reason why they have received considerable attention in recent years.
In the talk, some recent results regarding Rosenblatt processes will be given and an approach
to stochastic integration for Rosenblatt processes that is based on the Malliavin calculus will be
discussed. In particular, an Itô-type stochastic chain rule for processes with a second-order
fractional stochastic differential will be presented.
Rough volatility: SDE driven by Hölder continuous noise and unbounded
drift
Giulia Di Nunno, giulian@math.uio.no
University of Oslo, Norway
Coauthors: Yuliya Mishura, Anton Yurchenko-Tytarenko
Having in mind possible stochastic volatility models in finance, we consider an SDE driven by
a general Hölder continuous noise. The drift b is exploding with a control from below:
b(t, y) > CT (y − ϕ(t))−γ (γ > 0),
where ϕ is a continuous function and CT , γ are constants. We study the solution of such SDE
and its properties. In particular, we prove that it has a unique solution which is bound preserv-
ing.
Furthermore, modifying the controls on the drift, we obtain an SDE sandwiched between
two given bounds ϕ and ψ, with ψ(x) > ϕ(x). This sandwich solution turns out to be the most
useful for applications.
Among the properties presented, we show that the solution admits all moments. This re-
markable result paves the way for efficient numerical methods.
Exponential moments of hitting times for time-inhomogeneous atomic
Markov chains
Vitaliy Golomoziy, vitaliy.golomoziy@gmail.com
Taras Shevchenko National University of Kyiv, Ukraine
The result we present is devoted to studying exponential moments of hitting times for time-
inhomogeneous Markov chains. It is well-known that a necessary and sufficient condition for
555
