ニュース＆イベント

Math-Fi seminar on 21 Apr.

2022.04.20 Wed up
  • Date: 21 Apr. (Thu.)
  • Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
  • Time: 17:00-18:00
  • Speaker:  Jiro Akahori (Ritsumeikan University)
  • Title:  Variational approach to optimal stopping problems revisited
  • Abstract: 
After reviewing besoussan-lions’s variational approach, I will discuss its applications to numerical problems; discretization error, deep solver, and so on.  The talk will be in English.

Math-Fi seminar on 14 Apr.

2022.04.14 Thu up
  • Date: 14 Apr. (Thu.)
  • Place: On the Web
  • Time: 17:00-18:30
  • Speaker:  Umut Cetin (London School of Economics)
  • Title:  Speeding up the Euler scheme for killed diffusions
  • Abstract:
Let X be a linear diffusion taking values in  (l,r) and consider the standard Euler discretisation to compute the fair price of a Barrier option written on X that becomes worthless if X hits one of the barriers before the maturity date T. It is well-known since Gobet’s work that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to N^{-1/2} with N being the number of discretisatons. We introduce a drift-implicit Euler method to bring the convergence rate back to 1/N, i.e. the optimal rate in the absence of killing, using the theory of recurrent transformations. Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.
 

Math-Fi seminar on 7 Apr.

2022.04.11 Mon up
  • Date: 7 Apr. (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker:  Andrey Pilipenko (Ukraine National Academy of Sciences)
  • Title: Limit behavior of perturbed random walks
  • Abstract:
A particle moves randomly over the integer points of the real line. Jumps of the particle outside the membrane (a fixed “locally perturbating set”) are i.i.d., have zero mean and finite variance, whereas jumps of the particle from the membrane have other distributions with finite means which may be different for different points of the membrane; furthermore, these jumps are mutually independent and independent of the jumps outside the membrane. We prove that the weak scaling limit of the particle position is a skew Brownian motion.

立命館大学幾何学セミナー（2022年1月31日（月））

2022.01.25 Tue up
<<立命館大学幾何学セミナー>>

日時： 2022年1月31日（月） 16:30～18:00

タイトル： A geometric approach to stochastic extensions of nonholonomic constraints

講演者： François Gay-Balmaz（CNRS – LMD – Ecole Normale Supérieure）

アブストラクト：
We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange-d’Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals.
 
Inspired by this example, we then consider two different ways of including stochasticity in nonholonomic systems. We show that when the noise preserves the linearity of the constraints, then energy is.  preserved. For other types of noise in the constraint, e.g., in the case of an affine noise, the energy is not conserved. This approach is illustrated with a class of Lagrangian mechanical systems on Lie groups, with constraints of “rolling ball type”. We conclude with numerical simulations illustrating our theories, and some pedagogical examples of noise in constraints for other nonholonomic systems popular in the literature, such as the nonholonomic particle, the rolling disk and the Chaplygin sleigh.

開催方法：Zoom配信での開催です．

問い合わせ先：立命館大学理工学部数理科学科 多羅間 大輔

立命館大学幾何学セミナー（2022年1月24日（月））

2022.01.20 Thu up
<<立命館大学幾何学セミナー>>

日時：2022年1月24日（月） 15:00～16:00

タイトル：微分可能同相写像群の基本群

講演者： 前田 吉昭（東北大学）

アブストラクト：
微分可能同相写像群の基本群を調べるための不変量としてWodzicki-Chern-Simons クラスを定義する。この応用として、基本群が無限となる例として、シンプレクティック多様体上のサークル束について調べる。

開催方法：Zoom配信での開催です．

問い合わせ先：立命館大学理工学部数理科学科 多羅間 大輔

Math-Fi seminar on 20 Jan.

2022.01.19 Wed up
  • Date: 20 Jan. (Thu.)
  • Place: On the Web
  • Time: 16:30 – 18:00
  • Speaker: Stefano Pagliarani (University of Bologna)
  • Title: A Yosida’s parametrix approach to Varadhan’s estimates for a degenerate diffusion under the weak Hörmander condition
  • Abstract: 
We adapt and extend Yosida’s parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak H\”ormander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain a Varadhan-type formula for the asymptotic behavior of the logarithm of the transition density, in terms of the optimal cost function of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.

Math-Fi seminar on 13 Jan.

2022.01.12 Wed up
Date: 13 Jan. (Thu.)
Place: On the Web
Time: 16:30 – 18:00
Speaker: Benjamin Jourdain (CERMICS)
Title: Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q — L ρDrift Coefficient and Additive Noise
Abstract: 
We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q — L ρ drift coefficient when the condition d ρ + 2 q < 1, under which Krylov and R{ö}ckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 — (d ρ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density.

Math-Fi seminar on 6 Jan.

2022.01.06 Thu up
  • Date: 6 Jan. (Thu.)
  • Place: On the Web
  • Time: 18:00 – 19:30
  • Speaker: Xin Chen (Shanghai Jiao Tong University)
  • Title: Some results on backward stochastic differential equation on a Riemannian manifold
  • Abstract: 
In this talk, we will introduce some recent results on backward stochastic differential equation on a Riemannian manifold, including the definition of Riemannian-manifold valued BSDE, the probabilistic  representation for heat flow of harmonic map, the characterization of Navier-Stokes equation on a Riemannian manifold.
The talk is based on a joint work with Wenjie Ye.
 

立命館大学幾何学セミナー（2021年12月20日（月））

2021.12.09 Thu up
<<立命館大学幾何学セミナー>>

日時：2021年12月20日（月） 16:30～18:00

タイトル：
A bulk-edge correspondence through the second Chern number

講演者：
岩井 敏洋 （京都大学）

アブストラクト：
以下のURLからご覧ください．
http://www.math.ritsumei.ac.jp/home2/wp-content/uploads/geometry_seminar_poster_2021_12_20.pdf

開催方法：Zoom配信での開催です．

問い合わせ先：立命館大学理工学部数理科学科 多羅間 大輔

Math-Fi seminar on 9 Dec.

2021.12.09 Thu up
  • Date: 9 Dec. (Thu.)
  • Place: On the Web
  • Time: 16:30 – 18:00
  • Speaker: Takwa Saidaoui (University of Tunis El Manar) 
  • Title: Behavior of some discrete hedging errors in finance; a Fourier estimator in the presence of asynchronous trading
  • Abstract: 
This thesis focuses on three topics of financial mathematics. The first part consists of a study of the L^2-norm asymptotic behavior of the error due to the replicating portfolio discretization. The averaging feature of the Asian-type payoff plays a crucial role in improving the convergence rate of the error. We show that the achieved order is explicitly related to the fractional regularity of the payoff function. The second part studies the convergence rate of the error due to the discretization of the Clark-Ocone representation for functions of Levy processes with pure jumps. The obtained rate is strongly related to the regularity index of the Sobolev space to which the payoff belongs. The last part is a study of the asymptotic behavior (central limit theorem, CLT) of the Fourier estimator of the integrated covariance under the assumption of data asynchronicity. Thus, for a determinate choice of parameters, the estimator is consistent and the CLT is valid for a sub-optimal rate.
 

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