数理ファイナンスセミナー

Math-Fi seminar on 24 Sep.

2020.09.24 Thu up
  • Date: 24 Sep. (Thu.)
  • Place: On the Web
  • Time: 16:30 – 18:00
  • Speaker: Alex Mijatovic (University of Warwick)
  • Title: Invariance principle for non-homogeneous random walks
  • Abstract:
We discuss an invariance principle for a class of zero-drift spatially non-homogeneous random walks in ℝ^d, which may be recurrent in any dimension. The limit X is an elliptic martingale diffusion, which may be point-recurrent at the origin for any d≥2. To characterize X, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in ℝ^d and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of X and thus develop the excursion theory of X without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for X in ℝ^d, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of X is time-reversible. If so, the excursions of X in ℝ^d generalize the classical Pitman–Yor splitting-at-the-maximum property of Bessel excursions. This is joint work with N. Georgiou and A. Wade.

Math-Fi seminar on 17 Sep.

2020.09.17 Thu up
  • Date: 17 Sep. (Thu.) 
  • Place: On the Web
  • Time: 17:00 – 18:30 
  • Speaker: Stephane Menozzi
  • Title: Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift
 

Math-Fi seminar on 10 Sep.

2020.09.17 Thu up
  • Date: 10 Sep.  (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker: Ngo Hoang Long (Hanoi National University of Education)
  • Title: Tamed-adaptive Euler-Maruyama approximation for  SDEs with locally Lipschitz continuous drift and locally H\”older continuous diffusion coefficients
  • Abstract:
We propose a tamed-adaptive Euler-Maruyama approximation scheme for stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally H\”older continuous, polynomial growth diffusion coefficients. We consider the strong convergence and stability of the new scheme. In particular, we show that under some sufficient conditions for the stability of the exact solution,  the tamed-adaptive scheme converges strongly in any infinite time interval.
This is joint work with LUONG Duc Trong and KIEU Trung Thuy.

Math-Fi seminar on 3 Sep.

2020.09.03 Thu up
  • Date: 3 Sep. (Thu.)
  • Place: On the Web
  • Time: 16:30-18:00
  • Speaker: Linglong Yuan (University of Liverpool)
  • Title: Limit theorems for continuous-state branching processes with immigration
  • Abstract:
The continuous-state branching processes with immigration (CBI) arises in the literature of stochastic processes with a biological background.
Kawazu and Watanabe (1971) firstly introduced CBI and proved that it is the limit of a sequence of Galton-Waton discrete branching processes with immigration.
Since then CBI has received a lot of attention. It is connected to stochastic differential equations, Levy processes, population modelling etc. In mathematical finance, a special kind of CBI is called Cox–Ingersoll–Ross model which is well known to describe the evolution of interest rates. In this talk, we will briefly introduce CBI and focus on the long term behavior which turn out to have two different regimes. In the first one, an almost sure convergence is proved adapting the Grey’s martingale.
In the second one, only convergence in law is possible and a different technique is needed. Our results provide a global picture on the long term behavior and corrected a misprint in Pinsky’s paper [Bull. Amer. Math. Soc. 78 (1972), 242-244].
This talk is based on a joint work with Clement Foucart, Chunhua Ma.
 

Math-Fi seminar on 6 Aug.

2020.08.06 Thu up
  • Date: 6 Aug. (Thu.)
  • Place: On the Web 
  • Time: 16:30-18:00
  • Speaker: Alex Kulik (Wroclaw University of Science and Technology)
  • Title: Approximation in law of essentially singular Levy-type processes by non-linear regressions

Math-Fi seminar on 30 Jul.

2020.07.30 Thu up
  • Date: 30 Jul.  (Thu.)
  • Place: On the Web 
  • Time: 16:30-18:00
  • Speaker: Noufel Frikha (Paris VII)
  • Title: Yet another learning algorithm for Backward Stochastic Differential Equations
  • Abstract: 
​Backward Stochastic Differential Equations are stochastic processes that allow to represent the solution of semi-linear PDEs.
They are used to design numerical probabilistic methods for these PDEs. Recently, learning methods have proven to be successful in computing the solution of such equation in high dimensional settings.
We present a learning algorithm based on a stochastic gradient descent method and the use of sparse grids. We are able to prove precise rate of convergence for this numerical method, which allows to tame the curse of dimensionality in a smooth setting. We illustrate our theoretical results with some numerical experiments.
Joint work with J.-F. Chassagneux, J. Chen and C. Zhou.

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.
 

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