# Registered Data

## [01858] Interplay among Manifold Learning, Stochastic Calculus, and Volatility Estimation

**Session Date & Time**:- 01858 (1/3) : 2E (Aug.22, 17:40-19:20)
- 01858 (2/3) : 3C (Aug.23, 13:20-15:00)
- 01858 (3/3) : 3D (Aug.23, 15:30-17:10)

**Type**: Proposal of Minisymposium**Abstract**: We will review recent advances in Manifold Learning inviting top researchers in this field, and also invite some speakers who will deliver recent developments in Mallivin-Mancino's Fourier estimation method for estimating the "spot volatility process", and its application to the estimation of the diffusion matrix. The aim of the mini-symposium is to forecast how we can proceed to combine these two methods to refine the existing results and reach a new stage.**Organizer(s)**: Jiro Akahori, Hau-Tieng Wu**Classification**:__62G05__,__60F99__,__60H30__,__53A99__,__91G70__**Speakers Info**:- Ju-Yi Yen (University of Cincinatti)
- Hiroshi Kawabi (Keio University)
- Gi-Ren Liu (National Cheng Kung University)
- Chih-Wei Chen (National Sun Yat-sen University)
- Kei Kobayashi (Keio University)
- Masayuki Aino (Proxima Technology)
- Reika Kambara (Nomura Asset Management )
- Takatoshi Hirano (Misuho Bank)
- Riki KItano (Ritsumeikan University)

**Talks in Minisymposium**:**[02733] A graph discretized approximation of diffusions on Riemannian manifolds****Author(s)**:**Hiroshi Kawabi**(Keio University)- Satoshi Ishiwata (Yamagata University)

**Abstract**: In this talk, we discuss a graph discretized approximation for diffusions on a complete Riemannian manifold $M$. More precisely, for a given drifted Schrödinger operator ${\mathcal A}=-\Delta-b+V$ on $M$, we introduce a family of random walks in the flow generated by the drift $b$ with killing on a sequence of proximity graphs. The drifted Schrödinger semigroup $\{ {\rm e}^{t{\mathcal A}} \}_{t\geq 0}$ is approximated by discrete semigroups generated by the family of random walks.

**[02761] On excursions inside an excursion****Author(s)**:**Ju-Yi Yen**(University of Cincinnati)

**Abstract**: The distribution of ranked heights of excursions of a Brownian bridge is given by Pitman and Yor in $[1]$. In this work, we consider excursions of a Brownian excursion above a random level $x$, where $x$ is the value of the excursion at an independent uniform time on $[0,1]$. We analyze the maximum heights of these excursions as Pitman and Yor did for excursions of a Brownian bridge. $[1]$ J. Pitman and M. Yor. On the distribution of ranked heights of excursions of a Brownian bridge. Ann. Probab., 29(1):361–384, 2001.

**[02840] A Quantitative Central Limit Theorem arising from Time-Frequency Analysis****Author(s)**:**Gi-Ren Liu**(National Cheng Kung University)

**Abstract**: In this talk, we will discuss the distribution distance between the output $F$ of the scattering transform (ST) of a Gaussian process and its scaling limit $G$. ST is a nonlinear transformation that involves a sequential interlacing convolution and nonlinear operators, which is motivated to model the convolutional neural network. We will show that the total variation distance between the distributions of the output of ST and a chi-square random variable with one degree of freedom converges to zero at an exponential rate. For achieving this goal, we derive a recursive formula to represent the nonlinearity of ST by a linear combination of Wiener chaos and then apply the Malliavin calculus and Stein's method to estimate the maximal difference between the expectation values of $h(F)$ and $h(G)$ over a specific set of test functions $h$. This talk is based on joint work with Yuan-Chung Sheu (National Yang Ming Chiao Tung University, Taiwan) and Hau-Tieng Wu (Duke University, USA).

**[02841] Convergence of Hessian estimator from random samples on a manifold****Author(s)**:**Chih-Wei Chen**(National Sun Yat-sen University)- Hau-Tieng Wu (Duke University)

**Abstract**: We provide a systematic convergence analysis of the Hessian operator estimator from random samples supported on a low dimensional manifold. We show that the impact of the nonuniform sampling and the curvature on the widely applied Hessian operator estimator is asymptotically negligible.

**[02842] Diffusion Estimation with Fourier-Malliavin Method****Author(s)**:**Takatoshi Hirano**- Jiro Akahori (Ritsumeikan university)
- Simona Sanfelici (Parma university)

**Abstract**: In this talk, the estimation of the diffusion coefficient of the solution to a stochastic differential equation using Fourier estimation method proposed by P. Malliavin and M.E. Mancino is discussed.

**[02846] Market Price-Volatility Simulator****Author(s)**:**Riki Kitano**(Ritsumeikan University)

**Abstract**: In this talk, a combination of the Fourier-Malliavin-Mancino method and the Lyons’s signature method under a stochastic volatility diffusion setting will be discussed.

**[02848] Limit Theorems for the Positive Semidefinite Modification of Malliavin-Mancino Estimator for the Spot Volatility Process****Author(s)**:**Reika Kambara**(Ritsumeikan university)

**Abstract**: In this talk, the consistency, and the asymptotic normality of the class of Fourier-type estimators introduced by Akahori et al. will be discussed. The class, parameterized by a sequence of probability measures, is a modification of the Fourier series method introduced by Malliavin and Mancino, modified so that the estimator is positive semidefinite.

**[02849] Statistical Analysis with Geodesics and Curvature in Data Space****Author(s)**:**Kei Kobayashi**(Keio University)- Henry P. Wynn (London School of Economics)

**Abstract**: We proposed a method to perform data analysis after two types of metric transformation of the data space. The first transformation is based on the powered density integration and can be implemented approximately using an empirical graph. The second transformation corresponds to computing the extrinsic distance after embedding the data space into a metric cone. We proved both transformations monotonically change the curvature of the data space, but in different ways.

**[04489] Convergence of Laplacian and its rate for submanifolds that are not necessarily smooth****Author(s)**:**Masayuki Aino**(Proxima Technology)

**Abstract**: The continuous limit of Laplacian Eigenmaps gives the eigenfunctions of the Laplacian on submanifolds in Euclidean space . Such studies have been based on the assumption that a quantity called Reach is bounded from below. Submanifolds with non-differentiable points cannnot be approximated under such an assumption. In this talk, I discuss the convergence of Laplacian Eigenmaps and its rates when the Reach assumption is replaced by a weaker assumption such that non-differentiable points can appear.