立命館大学数理科学科 » 2021年度

Math-Fi seminar on 20 Jan.

horie — Wed, 19 Jan 2022 06:08:23 +0000

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.

horie — Wed, 12 Jan 2022 06:34:42 +0000

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.

horie — Thu, 06 Jan 2022 00:57:23 +0000

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.

Math-Fi seminar on 9 Dec.

horie — Thu, 09 Dec 2021 02:41:53 +0000

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.

Math-Fi seminar on 2 Dec.

horie — Wed, 01 Dec 2021 06:50:08 +0000

Date: 2 Dec. (Thu.)
Place: On the Web
Time: 16:30 – 18:00
Speaker: Ngoc Khue Tran ( Pham Van Dong University)

Math-Fi seminar on 25 Nov.

horie — Wed, 24 Nov 2021 07:33:22 +0000

Date: 25 Nov. (Thu.)
Place: On the Web
Time: 16:30 – 18:00
Speaker: Vlad Bally (University Paris Eiffel)
Title: Integration by parts and convergence in distribution norms in the CLT

Math-Fi seminar on 18 Nov.

horie — Thu, 18 Nov 2021 02:48:29 +0000

Date: 18 Nov. (Thu.)
Place: On the Web
Time: 16:30 – 18:00
Speaker: M2 students in Kohatsu lab. (Ritsumeikan University)

Math-Fi seminar on 21 Oct.

horie — Thu, 21 Oct 2021 02:32:15 +0000

Date: 21 Oct. (Thu.)
Place: On the Web
Time: 17:00 – 18:00
Speaker: Yasutaka Shimizu (Waseda University)
Title: A quite new approach to cohort-wise mortality prediction under survival energy hypothesis

Abstract:

We propose a new approach to mortality prediction by “Survival Energy Model (SEM)”．We assume that a human is born with initial energy, which changes stochastically in time and the human dies when the energy vanishes. Then, the time of death is represented by the first hitting time of the survival energy (SE) process to zero.

This study assumes that SE follows a (time-inhomogeneous) diffusion process or an inverse Gaussian process, and defines the “mortality function”, which is the first hitting time distribution function of a SE process. Although SEM is a fictitious construct, we illustrate that this assumption has a high potential to yield a good parametric family of the cumulative distribution of death, and the parametric family yields surprisingly good predictions for future mortality rates. This work is published by Shimizu, et al. (2020). “Why does a human die? A structural approach to cohort-wise mortality prediction under survival energy hypothesis”, ASTIN Bulletin, vol.51 (1), 191-219.

Math-Fi seminar on 14 Oct.

horie — Wed, 13 Oct 2021 08:24:48 +0000

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.

horie — Thu, 26 Aug 2021 00:34:27 +0000

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.