2021年度

Math-Fi seminar on 14 Oct.

2021.10.13 Wed up
• Date: 14 Oct. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Tomooki Yuasa (Ritsumeikan University)
• Title: Higher order error estimate of the discrete-time Clark–Ocone formula
• Abstract:
In this talk, we investigate the convergence rate of the discrete-time Clark–Ocone formula provided by Akahori–Amaba–Okuma (2017).
In that paper, they mainly focus on the $L_{2}$-convergence rate of the first order error estimate related to the tracking error of the delta hedge in mathematical finance.
Here, as two extensions, we estimate “the higher order error” for Wiener functionals with an integrability index $2$ and “an arbitrary differentiability index”.

Math-Fi seminar on 26 Aug.

2021.08.26 Thu up
• Date: 26 Aug. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Yuri Imamura (Kanazawa University)
• Title: Static Hedge via Parametrix and Symmetrization
• Abstract:
A scheme to construct an asymptotic expansion of static hedge of barrier options for multidimensional uniform elliptic diffusions leveraging both kernel symmetrization and parametrix techniques will be introduced.

Math-Fi seminar on 5 Aug.

2021.08.04 Wed up
• Date: 5 Aug. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Benjamin Poignard (Osaka University)
• Title: Estimation of High Dimensional Vector Autoregression via Sparse Precision Matrix
• Abstract:
We consider the problem of estimating sparse structural vector autoregression (SVAR) processes via penalized precision matrix. This matrix is the output of the underlying directed acyclic graph of the SVAR process, whose zero components correspond to zero SVAR coefficients. The precision matrix estimators are deduced from the class of Bregman divergences and regularized by the SCAD, MCP and LASSO penalties. Under suitable regularity conditions, we derive error bounds for the regularized precision matrix for each Bregman divergence. Moreover, we establish the support recovery property, including the case when the penalty is non-convex. These theoretical results are supported by empirical studies.

Math-Fi seminar on 29 Jul.

2021.07.28 Wed up
• Date: 29 Jul. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Takuya Nakagawa (Ritsumeikan University)
• Title: Projection scheme for polynomial diffusions on the unit ball
• Abstract:
In this talk, we consider numerical schemes for polynomial diffusions on the d-dimensional unit ball, which are solutions of stochastic differential equations with a diffusion coefficient of the form (1-|x|^{2})^{1/2}. We introduce a projection scheme on the unit ball based on a backward Euler–Maruyama scheme with the projection and provide the L^{2}-rate of convergence. The main idea to consider the numerical scheme is a transformation argument introduced by Swart, J. M. (2012) for proving the pathwise uniqueness for some stochastic differential equation with a non-Lipschitz diffusion coefficient. This study is a joint work with Dai Taguchi and Tomooki Yuasa.

Math-Fi seminar on 22 Jul.

2021.07.21 Wed up
• Date: 22 Jul. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Lorenzo Marino (Université d’Evry Val d’Essonne, University of Pavia)
• Title: Weak Regularization by Degenerate Lévy noise and Applications
• Abstract:
In this talk, we briefly present the arguments of the PhD thesis: “Weak Regularization by Degenerate Lévy noise and Applications”. After a general introduction on the regularization by noises phenomena and the motivations behind this work, we start by showing the Schauder estimates, a useful analytical tool for the wellposedness of SDEs, for two different classes of integro-differential equations whose coefficients lie in suitable anisotropic Hölder spaces with multi-indices of regularity. The first one focuses on non-linear dynamics controlled by an α-stable operator acting only on the first component. To deal with the non-linear perturbation, we also need some subtle controls on Besov norms. As an extension of the first one, we also present the Schauder estimates associated with a degenerate Ornstein-Uhlenbeck operator driven by a larger class of α-stabletype operators, like the relativistic or Lamperti stable ones. The proof of this result relies instead on a precise analysis of the behaviour of the associated Markov semigroup between anisotropic Hölder spaces and some interpolation techniques. Exploiting a backward parametrix approach, we finally prove the weak wellposedness of the associated degenerate chain of SDEs. As a by-product of our method, Krylov-type estimates on the canonical solution process are also presented. Time permitting, we conclude by showing through suitable counter-examples that there exists an (almost) sharp threshold for the regularity exponents that ensure the weak well-posedness for the SDE.

Math-Fi seminar on 8 Jul.

2021.07.07 Wed up
• Date: 8 Jul. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Shota Nakamura (Waseda University)
• Title: 長期記憶性を持つサープラスの破産確率の推定
• Abstract:
クーレムが長期記憶性を持つような保険会社のサープラスの推定問題を考える.
この様なモデルは近似的にハースト指数H>1/2を持つ非整数ブラウン運動で表現することができる. 本セミナーでは,非整数ブラウン運動で表現されるサープラスのモデルが持つ未知パラメータの漸近正規性を持つ推定量が与えられた際の,破産確率の推定量の漸近分布のMalliavin解析を用いた導出法について概説を行う.

Math-Fi seminar on 1 Jul.

2021.06.30 Wed up
• Date: 1 Jul. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Roger Pettersson (Linnaeus University)
• Title: Epidemic modeling on microscopic, macroscopic and mesoscopic scale: the Kurtz’ approach

Math-Fi seminar on 24 Jun.

2021.06.23 Wed up
• Date: 24 Jun. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Maria Elvira Mancino (University of Florence)
• Title: Volatility and higher order covariances estimation for identifying financial instability conditions and computing hedging Greeks
• Abstract:
I would present the most recent papers with Simona, both of them use the same mathematical instruments for different applications. Further, it is based on some ideas presented in the paper with Malliavin  “”Harmonic analysis methods for nonparametric estimation of volatility: theory and applications””. In  Proceedings of the International Symposium “Stochastic Processes and Applications to Mathematical Finance” 2005 at Ritsumeikan University, World Scientific (2006).  Eds. J.Akahori, S.Ogawa, S.Watanabe.

Math-Fi seminar on 17 Jun.

2021.06.17 Thu up
• Date: 17 Jun. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Olivier Menoukeu Pamen (African Institute of Mathematical Sciences and University of Liverpool)
• Title:Takagi type functions and dynamical systems: the smoothness of the SBR measure and the existence of local time

Math-Fi seminar on 10 Jun.

2021.06.09 Wed up
• Date: 10 Jun. (Thu.)
• Place: On the Web
• Time: 16:30 – 18:00
• Speaker: Pierre Bras (Sorbonne University, LPSM)
• Title: Convergence rates of Gibbs measures with degenerate minimum
• Abstract:
We study convergence rates of Gibbs measures, with density $\pi_t(dx) \propto e^{-f(x)/t} dx$, as $t \to 0$ and where $f: \mathbb{R}^d \to \mathbb{R}$ admits a unique global minimum at $x^\star$. If the Hessian matrix $\nabla^2 f(x^\star)$ is positive definite then a Taylor expansion up to order 2 shows that $\pi_t$ converges to the Dirac measure $\delta_{x^\star}$ at speed $\sqrt{t}$.
We focus on the case where the Hessian of $f$ is not definite at $x^\star$. We assume instead that the minimum is strictly polynomial and we give a higher order nested expansion of $f$ at $x^\star$. We give an algorithm yielding such decomposition, in connection with Hilbert’s $17^{th}$ problem. We then give the rate of convergence of $\pi_t$ using this expansion.
Our work can be applied to stochastic optimization, where the Gibbs measure $\pi_t$ with small $t$ is used as an approximation of the minimizer of $f$.