2025年度

Math-Fi seminar on 15 May.

2025.05.13 Tue up
  • Date: 15 May. (Thu.) 
  • Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
  • Time:  16:50–19:00
     
  • Speaker 1: Ju-Yi Yen (University of Cincinnatti)
  • Time: 16:50–17:50
  • Title: Excursion-Theoretic Approaches to Limit Theorems for Additive Functionals of Markov Processe
  • Abstract:
This talk explores a uni ed excursion-theoretic framework for proving limit theorems of additive functionals associated with various classes of Markov processes, including Brownian motion, null recurrent di usions, and symmetric strong Markov processes. Motivated by classical results such as the Darling Kac theorem and the Ray Knight theorems, we investigate how local time provides a natural time scale for analyzing functionals of processes that do not admit nite invariant measures. We begin with Brownian motion, where the inverse local time at zero enables a strong law of large numbers and central limit theorem for time integrals of functions along sample paths. These results are revisited using Itos excursion theory, highlighting its utility in both deriving moments and capturing uctuation behavior. We then extend these ideas to null recurrent linear di usions by transforming them into time-changed Brownian motions via Zvonkins method. Excursion-based representations again yield central limit theorems, even in the absence of stationary distributions. Finally, we examine a broader setting of symmetric strong Markov processes, where local times are used to de ne regenerative structures. By leveraging generalized RayKnight theorems and Gaussian process techniques, we establish limit theorems under minimal assumptions, unifying previous results under a single probabilistic strategy. This excursion-centric viewpoint not only clari es asymptotic behaviors but also opens paths toward analyzing more complex dynamics such as complex-valued processes and higher-dimensional extensions.

 
  • Speaker 2: Loïc Chaumont (Université d’Angers)
  • Time: 18:00–19:00
  • Title: Levy processes resurrected in the positive half-line
  • Abstract:
Levy processes resurrected in the positive half-line is a Markov process obtained by removing successively all jumps that make it negative. A natural question, given this construction, is whether the resulting process is absorbed at 0 or not. In this work, we give conditions for absorption and conditions for non absorption bearing on the characteristics of the initial Levy process. First, we shall give a detailed definition of the resurrected process whose law is described in terms of that of the process killed when it reaches the negative half line. In particular, we will specify the explicit form of the resurrection kernel. Then we will see that when the initial Levy process X creeps downward and satisfies certain additional condition, the resurrected process is absorbed at 0 with probability one, independently of its starting
point. Some criteria for absorption and some criteria for non absorption will be given. The most delicate case is when X enters immediately in the negative half line and drifts to -infinity. It is then possible to give a sufficient condition for absorption but up to now, even when X is the negative of a subordinator, we do not know whether this condition can be dropped or not. We shall take a closer look at the case of stable processes. This is a joint work with Victor Rivero (CIMAT, Guanajuato) and Marria Emilia Caballero (Instituto de Matematicas, UNAM, Mexico).
 

Math-Fi seminar on 24 Apr.

2025.04.22 Tue up
  • Date: 24 Apr. (Thu.) 
  • Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
  • Time: 13:10–19:00


  • Speaker 1: Dima Ivanenko (Taras Shevchenko National University of Kyiv)
  • Time: 13:10–14:10
  • Title: ON APPROXIMATION OF SOME LÉVY PROCESSES <4>
  • Abstract:
A Levy process X(t) has the structure X(t) = at + σW(t) + J(t) where W(t) is standard
Brownian motion (BM) and J(t) an independent pure jump process. This class of processes
has been used in numerous application areas, of which we in particular mention nance
and queueing. Calculations for a Levy process are, however, in general more dicult than
for BM, and an abundance of expressions that are explicit for BM are not so even in the
most popular parametric Levy models. Simulation of X(t) is therefore one of the main
computational tools.
A Levy process has countably many jumps on any interval [0, T] and nitely many jumps
of size bigger than some xed ε > 0. In order to simulate Z, we need to take nitely many
jumps of Z, which gives an adequate description of Z. Apart from some particular cases,
e.g. Brownian motion, Gamma process, α-stable process, simulation of the Levy process
with a given triplet is not an easy task.
Usually, the distribution function of a Levy process is unknown or has a rather compli-
cated form, which makes the simulation rather perplex. For the methods of generating
innitely divisible random variables (r.v.’s) and Levy processes (e.g. methods of Khinchin,
Fergusson-Klass, Bondesson, LePage, Rosinski) we refer to Rosinski (2001) and propose
our own method. We also would like to mention that the Damien-Laud-Smith algorithm
from Damien, Laud, and Smith (1995) gives a way to simulate an (approximation of)
an arbitrary onedimensional innitely divisible r.v., which allows us to simulate a Levy
process. On the other hand, it was observed by Bondesson (1982) and later by Asmussen
and Rosinski (2001), that under some conditions small jumps can be substituted by an
(arithmetic) Brownian motion.
The series of seminars includes a general theory of Levy processes, an overview of known
methods for modeling such processes, and a comparison of these methods with our own.
 
 
  • Speaker 2: Oleksii Kulik　 (Wroclaw University of Science and Technology)
  • Time: 14:20-15:20
  • Title: A moments respecting explicit simulation scheme for L´evy driven SDEs, II: Simulation methods for SDEs with super-linearly dissipative drifts.
  • Abstract:
This is the first of two lectures devoted to the effect of tails/moments transformation by a dissipative drift for  solutions of SDEs driven by heavy-tailed noises.  In the first lecture we will discuss the history of the subject, illustrate how does this effect reveals itself in various settings, and provide a complete description of the effect in a general semi-martingale setting.
 
 
  • Speaker 3: Maria Elvira Mancino (University of Florence)
  • Time: 16:50-17:50
  • Title: Asymptotic Efficiency of the Fourier Spot Volatility Estimator with Noisy Data and an Application to the estimation of spot Beta and higher-order spot covariances.
  • Abstract:
We study the efficiency and robustness of the Fourier spot volatility estimator when high-frequency prices are contaminated by microstructure noise. Firstly, we show that the estimator is consistent and asymptotically efficient in  the presence of additive noise, establishing a Central Limit Theorem with the optimal rate of convergence 1/8. Feasible CLTs with the optimal convergence rate are also obtained, by proving the consistency of the Fourier estimators of the spot volatility of volatility and the quarticity in the presence of noise. The multivariate case is also studied.
Finally, we exploit this methodology to introduce a consistent estimator of the spot asset beta. We provide simulation results that suggest that our spot beta estimator has a robust finite-sample performance in the presence of realistic market features such as rough volatility, inhomogeneous asynchronous sampling, autocorrelated and price-dependent noise and price rounding. Additionally, we conduct an empirical study with tick-by-tick prices in which we reconstruct intraday spot beta paths.
 
 
  • Speaker 4: Shigeki Matsutani（Kanazawa University）
  • Time: 18:00-19:00
  • Title: 量子ウォークと光学
  • Abstract:
本講演では，一次元量子ウォークと光学におけるS行列理論との関連を示す．
光学における転送行列理論は，一次元のHermholz方程式を離散化することで得られる．
この転送行列の系を入射光と出射光で再定式化すると，S行列（散乱行列）が得られる．
さらに，入射光から出射光への静的な変換を，因果関係あるいは一種の離散的時間発展として動的な波動系を考えると，1次元量子ウォークと一致する．
産業界における光学現象には，波動系の根幹に関わる基本的な問題が非常に多い．
我々は，この量子ウォークを光学系として解釈することより，波動力学の本質と基礎が得られると考えている．
例えば，ファインマンの著書「光と物質のふしぎな理論」の描像に従い，量子ウォークはフェルマーの原理を満たし，波面は光速で弾道的に進むと見ることができる．
さらに，量子ウォークの振る舞いは，ド・ブロイ・ボーム理論のパイロット波を想起する結果を得る．

Math-Fi seminar on 17 Apr.

2025.04.15 Tue up
  • Date: 17 Apr. (Thu.) 
  • Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
  • Time: 16:50–19:00 


  • Speaker 1: Dima Ivanenko (Taras Shevchenko National University of Kyiv)
  • Time: 16:50–17:50 
  • Title: ON APPROXIMATION OF SOME LÉVY PROCESSES 3
  • Abstract:
A Levy process X(t) has the structure X(t) = at + σW(t) + J(t) where W(t) is standard
Brownian motion (BM) and J(t) an independent pure jump process. This class of processes
has been used in numerous application areas, of which we in particular mention nance
and queueing. Calculations for a Levy process are, however, in general more dicult than
for BM, and an abundance of expressions that are explicit for BM are not so even in the
most popular parametric Levy models. Simulation of X(t) is therefore one of the main
computational tools.
A Levy process has countably many jumps on any interval [0, T] and nitely many jumps
of size bigger than some xed ε > 0. In order to simulate Z, we need to take nitely many
jumps of Z, which gives an adequate description of Z. Apart from some particular cases,
e.g. Brownian motion, Gamma process, α-stable process, simulation of the Levy process
with a given triplet is not an easy task.
Usually, the distribution function of a Levy process is unknown or has a rather compli-
cated form, which makes the simulation rather perplex. For the methods of generating
innitely divisible random variables (r.v.’s) and Levy processes (e.g. methods of Khinchin,
Fergusson-Klass, Bondesson, LePage, Rosinski) we refer to Rosinski (2001) and propose
our own method. We also would like to mention that the Damien-Laud-Smith algorithm
from Damien, Laud, and Smith (1995) gives a way to simulate an (approximation of)
an arbitrary onedimensional innitely divisible r.v., which allows us to simulate a Levy
process. On the other hand, it was observed by Bondesson (1982) and later by Asmussen
and Rosinski (2001), that under some conditions small jumps can be substituted by an
(arithmetic) Brownian motion.
The series of seminars includes a general theory of Levy processes, an overview of known
methods for modeling such processes, and a comparison of these methods with our own.
 
  • Speaker 2: Oleksii Kulik　 (Wroclaw University of Science and Technology)
  • Time: 18:00–19:00 
  • Title: A moments respecting explicit simulation scheme for L\’evy driven SDEs, I: Transformation of a heavy-tailed L\’evy noise by a dissipative drift
  • Abstract:
This is the first of two lectures devoted to the effect of tails/moments transformation by a dissipative drift for  solutions of SDEs driven by heavy-tailed noises.  In the first lecture we will discuss the history of the subject, illustrate how does this effect reveals itself in various settings, and provide a complete description of the effect in a general semi-martingale setting.

Math-Fi seminar on 10 Apr. (Co-organized as a Quantum Walk Seminar)

2025.04.08 Tue up
  • Date: 10 Apr. (Thu.) 
  • Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
  • Time: 16:50–17:50 
  • Speaker 1: Dima Ivanenko (Taras Shevchenko National University of Kyiv)
  • Title: ON APPROXIMATION OF SOME LÉVY PROCESSES 2
  • Abstract:
A Levy process X(t) has the structure X(t) = at + σW(t) + J(t) where W(t) is standard
Brownian motion (BM) and J(t) an independent pure jump process. This class of processes
has been used in numerous application areas, of which we in particular mention nance
and queueing. Calculations for a Levy process are, however, in general more dicult than
for BM, and an abundance of expressions that are explicit for BM are not so even in the
most popular parametric Levy models. Simulation of X(t) is therefore one of the main
computational tools.
A Levy process has countably many jumps on any interval [0, T] and nitely many jumps
of size bigger than some xed ε > 0. In order to simulate Z, we need to take nitely many
jumps of Z, which gives an adequate description of Z. Apart from some particular cases,
e.g. Brownian motion, Gamma process, α-stable process, simulation of the Levy process
with a given triplet is not an easy task.
Usually, the distribution function of a Levy process is unknown or has a rather compli-
cated form, which makes the simulation rather perplex. For the methods of generating
innitely divisible random variables (r.v.’s) and Levy processes (e.g. methods of Khinchin,
Fergusson-Klass, Bondesson, LePage, Rosinski) we refer to Rosinski (2001) and propose
our own method. We also would like to mention that the Damien-Laud-Smith algorithm
from Damien, Laud, and Smith (1995) gives a way to simulate an (approximation of)
an arbitrary onedimensional innitely divisible r.v., which allows us to simulate a Levy
process. On the other hand, it was observed by Bondesson (1982) and later by Asmussen
and Rosinski (2001), that under some conditions small jumps can be substituted by an
(arithmetic) Brownian motion.
The series of seminars includes a general theory of Levy processes, an overview of known
methods for modeling such processes, and a comparison of these methods with our own.
 
  • Speaker 2: Sohei Tateno (Nagoya University)
  • Time: 18:00–19:00 
  • Title: Iwasawa theory for discrete-time quantum walks in graphs
  • Abstract:
Transition matrices of discrete-time quantum walks in graphs are closely related to Ihara zeta functions of weighted graphs. For example, the celebrated Konno—Sato theorem has an application to calculate the eigenvalues of transition matrices. In this talk, by applying the methods of Iwasawa theory for graphs, we establish an asymptotic formula for the values of the characteristic polynomials of the transition matrices in a $\mathbb{Z}_p^d$-tower of graphs when an element of $\overline{\mathbb{Q}}_p$ is substituted to the polynomials. This is a joint work with Taiga Adachi and Kosuke Mizuno. 

Math-Fi seminar on 4 Apr.

2025.04.04 Fri up
  • Date: 4 Apr. (Fri.) 
     
  • Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
     
  • Time: 16:30–18:00
     
  • Speaker:  Dima Ivanenko (Taras Shevchenko National University of Kyiv)
     
  • Title: ON APPROXIMATION OF SOME LÉVY PROCESSES
     
  • Abstract:
A Levy process X(t) has the structure X(t) = at + σW(t) + J(t) where W(t) is standard
Brownian motion (BM) and J(t) an independent pure jump process. This class of processes
has been used in numerous application areas, of which we in particular mention nance
and queueing. Calculations for a Levy process are, however, in general more dicult than
for BM, and an abundance of expressions that are explicit for BM are not so even in the
most popular parametric Levy models. Simulation of X(t) is therefore one of the main
computational tools.
A Levy process has countably many jumps on any interval [0, T] and nitely many jumps
of size bigger than some xed ε > 0. In order to simulate Z, we need to take nitely many
jumps of Z, which gives an adequate description of Z. Apart from some particular cases,
e.g. Brownian motion, Gamma process, α-stable process, simulation of the Levy process
with a given triplet is not an easy task.
Usually, the distribution function of a Levy process is unknown or has a rather compli-
cated form, which makes the simulation rather perplex. For the methods of generating
innitely divisible random variables (r.v.’s) and Levy processes (e.g. methods of Khinchin,
Fergusson-Klass, Bondesson, LePage, Rosinski) we refer to Rosinski (2001) and propose
our own method. We also would like to mention that the Damien-Laud-Smith algorithm
from Damien, Laud, and Smith (1995) gives a way to simulate an (approximation of)
an arbitrary onedimensional innitely divisible r.v., which allows us to simulate a Levy
process. On the other hand, it was observed by Bondesson (1982) and later by Asmussen
and Rosinski (2001), that under some conditions small jumps can be substituted by an
(arithmetic) Brownian motion.
The series of seminars includes a general theory of Levy processes, an overview of known
methods for modeling such processes, and a comparison of these methods with our own.