Seminars

Math-Fi seminar on 23 Jul.

2020.07.23 Thu up
  • Date: 23 Jul.  (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker: Toshihiro Yamada (Hitotsubashi University)
  • Title: Higher order weak approximation for SDEs and BSDEs of McKean-Vlasov type
  • Abstract:
In this talk, we give a higher order discretization method for McKean-Vlasov type SDEs. Numerical examples for the discretization (up to fourth order) scheme are shown for McKean-Vlasov SDEs. Also, we introduce a second order discretization scheme for decoupled McKean-Vlasov BSDEs. Some applications will be discussed.
The talk is based on the joint work with Riu Naito.

Math-Fi seminar on 16 Jul.

2020.07.16 Thu up
  • Date: 16 Jul.  (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker: Fabio Antonelli (University of L’Aquila)
  • Title: Evaluation via power expansion

Math-Fi seminar on 9 Jul.

2020.07.09 Thu up
  • Date: 9 Jul. (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker: Noufel Frikha (Université de Paris)
  • Title: Well-posedness of McKean-Vlasov SDEs, related PDE on the Wasserstein space and some new quantitative estimates for propagation of chaos.
  • Abstract:
In this talk, i will present some new well-posedness results for non-linear diffusion/jump processes in the sense of McKean-Vlasov which go beyond the (well-understood) Cauchy-Lipschitz theory (see e.g. the course at St-Flou of A.S. Sznitman). For non-linear diffusion processes, I will show how the underlying noise regularizes the system and allows to establish the existence and the regularity properties of the transition density with respect to the measure argument, under a uniform ellipticity assumption. I will present the link between this smoothing effect with respect to the initial measure and the Backward Kolmogorov PDE on the Wasserstein space, which is the space of probability measures with finite second order moment. Finally, I will show how classical solutions to this PDE play a key role to establish some new quantitative estimates of propagation of chaos for the approximation of the mean-field dynamics by the related particle system.
This presentation is based on some recent works in collaboration with: P.-E. Chaudru de Raynal (Université Savoie Mont Blanc), V. Konakov (HSE Moscou), L. Li (UNSW Sydney) and S. Menozzi (Université d’Evry Val d’Essone).
 

Math-Fi seminar on 2 Jul.

2020.07.02 Thu up
  • Date: 2 Jul. (Thu.)
  • Place: On the web
  • Time: 16:30-18:00
  • Speaker: Dan Crisan (Imperial College London)
  • Title: On stochastic partial differential equations driven by transport noise
  • Abstract: 
I will discuss some (local and global) well-posedness results and a Beale–Kato–Majda blow-up criterion for stochastic fluid equations for incompressible flows. This is joint work with Franco Flandoli, Darryl Holm, Oana Lang and Oliver Street. The work is motivated by application to the study of upper ocean dynamics.
 

Math-Fi seminar on 18 Jun.

2020.06.18 Thu up
  • Date: 18 Jun.  (Thu.)
  • Place: On the Web
  • Time: 16:00-17:00
  • Speaker: Arturo Kohatsu-Higa (Ritsumeikan) 
  • Title: 信頼空間、Light TailかFat Tailか？
 

Math-Fi seminar on 4 Jun.

2020.06.04 Thu up
  • Date: 4 Jun.  (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker: Guanting Liu (UNSW)
  • Title: A positivity-preserving numerical scheme for the alpha-CEV process
  • Abstract: 
​We propose and prove strong convergence of a positivity-preserving implicit numerical scheme for jump-extended Cox-Ingersoll-Ross (CIR) process and Constant-Elasticity-of-Variance (CEV) process, where the jumps are governed by a compensated spectrally positive alpha-stable Levy process for alpha in (1, 2).
This class of models have first been studied in the context of continuous branching processes with interaction and/or immigration, and in this class a model has been introduced to mathematical finance for modelling sovereign interest rates and the energy market, which was named the alpha-CIR process. Numerical schemes for jump-extended CIR and CEV processes in the current literature, to the best of our knowledge, have all focused on the case of finite activity jumps (e.g. Poisson jumps) except our previous work studying a positivity-preserving scheme for the alpha-CIR process. In this paper, besides strong convergence we also obtain bounded beta-moments of the numerical scheme, for beta in [1, alpha), which allows us to left the boundedness requirement on the jump coefficient, and hence avoid truncation.
 

Math-Fi seminar on 28 May

2020.05.28 Thu up
Date: 28 May (Thu.)
Place: On the Web
Time: 16:30-18:00
Speaker: Takafumi Amaba (Fukuoka University)
Title: Bayesian CNNとTsirelson-Vershikによる黒色雑音の構成について
 

Math-Fi seminar on 7 May

2020.05.07 Thu up
  • Date: 7 May (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker: Takuya Nakagawa (Ritsumeikan University)
  • Title: On a Monte Carlo scheme for a stochasticquantity of SPDEs with discontinuous initial conditions
  • Abstract:
The aim of this seminar is to study the simulation of an expectation of a stochastic quantity $\e[f(u(t,x))]$ for a solution of stochastic partial differential equation driven by multiplicative noise with a non-smooth coefficients and a boundary condition: $Lu(t,x)=h(t,x) \dot{W}(t,x)$.
We first define a Monte Carlo scheme $P_{t}^{(N,M,L)}f(x)$ for $P_{t}f(x):=\e[f(u(t,x))]$, where $f$ is a bounded measurable function $f$ and $u(t,x)$ is a solution of stochastic partial differential equation given by Duhamel’s formula, and then we prove the convergence of the Monte Carlo scheme $P_{t}^{(N,M,L)}f(x)$ to $P_{t}f(x)$ and  the rate of weak error.
In addition, we introduce results of numerical experiments about the convergence error and the Central limit theorem for the scheme.

Math-Fi seminar on 23 Apr.

2020.04.23 Thu up
  • Date:  23 Apr. (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker: Dai Taguchi (Okayama University)
  • Title: Multi-dimensional Avikainen’s estimates
  • Abstract:
Avikainen provided a sharp upper bound of the expectation of |f(X)-f(Y)|^{q} by the expectation of |X-Y|^{p}, for any one-dimensional random variables X with bounded density and Y, and function of bounded variation f. In this talk, we consider multi-dimensional analogues of this estimate for any function of bounded variation in R^{d}, Orlicz–Sobolev spaces, Sobolev spaces with variable exponents or fractional Sobolev spaces.

We apply main statements to the numerical analysis on irregular functional of a solution to stochastic differential equations based on the Euler–Maruyama scheme and the multilevel Monte Carlo method, and L^{2}-time regularity of decoupled forward–backward stochastic differential equations with irregular terminal conditions.

This is joint work with Akihiro Tanaka (Osaka university) and Tomooki Yuasa (Ritsumeikan University).

Math-Fi seminar on 9 Apr.

2020.04.09 Thu up
  • Date: 9 Apr. (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker: Pierre Bras (École normale supérieure Paris)
  • Title: Acceleration of stochastic optimization algorithms
  • abstract: 
I will introduce stochastic algorithms to solve optimization problems, arising in machine learning or mathematical finance. Then, starting with the basic stochastic gradient algorithm, we will see how one can modify it so that to accelerate the convergence towards the optimal solution.
 

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