## 統計数学セミナー

過去の記録 ～02/25｜次回の予定｜今後の予定 02/26～

担当者 | 吉田朋広、荻原哲平、小池祐太 |
---|---|

セミナーURL | http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/ |

目的 | 確率統計学およびその関連領域に関する研究発表, 研究紹介を行う. |

**過去の記録**

### 2017年11月02日(木)

14:00-15:10 数理科学研究科棟(駒場) 052号室

Hermite processes and sheets

**Tudor Ciprian 氏**(Université Lille 1)Hermite processes and sheets

[ 講演概要 ]

The Hermite process of order $\geq 1$ is a self-similar stochastic process with stationary increments living in the $q$th Wiener chaos. The class of Hermite processes includes the fractional Brownian motion (for $q=1$) and the Rosenblatt process (for $q=2$). We present the basic properties of these processes and we introduce their multiparameter version. We also discuss the behavior with respect to the self-similarity index and the possibility so solve stochastic equations with Hermite noise.

The Hermite process of order $\geq 1$ is a self-similar stochastic process with stationary increments living in the $q$th Wiener chaos. The class of Hermite processes includes the fractional Brownian motion (for $q=1$) and the Rosenblatt process (for $q=2$). We present the basic properties of these processes and we introduce their multiparameter version. We also discuss the behavior with respect to the self-similarity index and the possibility so solve stochastic equations with Hermite noise.

### 2017年08月23日(水)

13:30-14:40 数理科学研究科棟(駒場) 052号室

Covariation estimation from noisy Gaussian observations:equivalence, efficiency and estimation

**Sebastian Holtz 氏**(Humboldt University of Berlin)Covariation estimation from noisy Gaussian observations:equivalence, efficiency and estimation

[ 講演概要 ]

In this work the estimation of functionals of the quadratic covariation matrix from a discretely observed Gaussian path on [0,1] under noise is discussed and analysed on a large scale. At first asymptotic equivalence in Le Cam's sense is established to link the initial high-frequency model to its continuous counterpart. Then sharp asymptotic lower bounds for a general class of parametric basic case models, including the fractional Brownian motion, are derived. These bounds are generalised to the nonparametric and even random parameter setup for certain special cases, e.g. Itô processes. Finally, regular sequences of spectral estimators are constructed that obey the derived efficiency statements.

In this work the estimation of functionals of the quadratic covariation matrix from a discretely observed Gaussian path on [0,1] under noise is discussed and analysed on a large scale. At first asymptotic equivalence in Le Cam's sense is established to link the initial high-frequency model to its continuous counterpart. Then sharp asymptotic lower bounds for a general class of parametric basic case models, including the fractional Brownian motion, are derived. These bounds are generalised to the nonparametric and even random parameter setup for certain special cases, e.g. Itô processes. Finally, regular sequences of spectral estimators are constructed that obey the derived efficiency statements.

### 2017年05月18日(木)

15:00-16:10 数理科学研究科棟(駒場) 117号室

On a representation of fractional Brownian motion and the limit distributions of statistics arising in cusp statistical models

**Alexander A. Novikov 氏**(University of Technology Sydney)On a representation of fractional Brownian motion and the limit distributions of statistics arising in cusp statistical models

[ 講演概要 ]

We discuss some extensions of results from the recent paper by Chernoyarov et al. (Ann. Inst. Stat. Math. October 2016) concerning limit distributions of Bayesian and maximum likelihood estimators in the model "signal plus white noise" with irregular cusp-type signals. Using a new representation of fractional Brownian motion (fBm) in terms of cusp functions we show that as the noise intensity tends to zero, the limit distributions are expressed in terms of fBm for the full range of asymmetric cusp-type signals correspondingly with the Hurst parameter H, 0＜H＜1. Simulation results for the densities and variances of the limit distributions of Bayesian and maximum likelihood estimators are also provided.

We discuss some extensions of results from the recent paper by Chernoyarov et al. (Ann. Inst. Stat. Math. October 2016) concerning limit distributions of Bayesian and maximum likelihood estimators in the model "signal plus white noise" with irregular cusp-type signals. Using a new representation of fractional Brownian motion (fBm) in terms of cusp functions we show that as the noise intensity tends to zero, the limit distributions are expressed in terms of fBm for the full range of asymmetric cusp-type signals correspondingly with the Hurst parameter H, 0＜H＜1. Simulation results for the densities and variances of the limit distributions of Bayesian and maximum likelihood estimators are also provided.

### 2017年04月20日(木)

15:00- 数理科学研究科棟(駒場) 117号室

Central limit theorem for symmetric integrals

Stochastic heat equation with rough multiplicative noise

**David Nualart 氏**(Kansas University) -Central limit theorem for symmetric integrals

[ 講演概要 ]

The purpose of this talk is to present the convergence in distribution of symmetric integrals of functions of the fractional Brownian motion for critical values of the Hurst parameter. This result includes the cases of symmetric integrals defined as the limit of trapeziodal, midpoint and Simpson Riemann sums, where the corresponding critical values of the Hurst parameter are H=1/4, H=1/6 and H=1/10, respectively. As a consequence, we establish a change-of-variable formula in law, where the correction term involves a stochastic integral with respect to an independent standard Brownian motion. The proof is based on the combination of Malliavin calculus and the classical Bernstein's big blocks/small blocks technique.

The purpose of this talk is to present the convergence in distribution of symmetric integrals of functions of the fractional Brownian motion for critical values of the Hurst parameter. This result includes the cases of symmetric integrals defined as the limit of trapeziodal, midpoint and Simpson Riemann sums, where the corresponding critical values of the Hurst parameter are H=1/4, H=1/6 and H=1/10, respectively. As a consequence, we establish a change-of-variable formula in law, where the correction term involves a stochastic integral with respect to an independent standard Brownian motion. The proof is based on the combination of Malliavin calculus and the classical Bernstein's big blocks/small blocks technique.

**David Nualart 氏**(Kansas University) -Stochastic heat equation with rough multiplicative noise

[ 講演概要 ]

The aim of this talk is to present some results on the existence and uniqueness of a solution for the one-dimensional heat equation driven by a Gaussian noise which is white in time and it has the covariance of a fractional Brownian motion with Hurst parameter less than 1/2 in the space variable. In the linear case we establish a Feynman-Kac formula for the moments of the solution and discuss intermittency properties.

The aim of this talk is to present some results on the existence and uniqueness of a solution for the one-dimensional heat equation driven by a Gaussian noise which is white in time and it has the covariance of a fractional Brownian motion with Hurst parameter less than 1/2 in the space variable. In the linear case we establish a Feynman-Kac formula for the moments of the solution and discuss intermittency properties.

### 2017年03月07日(火)

14:00-15:30 数理科学研究科棟(駒場) 052号室

大阪大学基礎工学研究科棟 I407号室 (WEB配信）

Nonparametric change-point analysis of volatility

大阪大学基礎工学研究科棟 I407号室 (WEB配信）

**Markus Bibinger 氏**(Humboldt-Universität zu Berlin)Nonparametric change-point analysis of volatility

[ 講演概要 ]

We develop change-point methods for statistics of high-frequency data. The main interest is in the stochastic volatility process of an Itô semi-martingale, the latter being discretely observed over a fixed time horizon. For a local change-point problem under high-frequency asymptotics, we construct a minimax-optimal test to discriminate continuous volatility paths from paths comprising changes. The key example is identification of volatility jumps. We prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. Moreover, we study a different global change-point problem to identify changes in the regularity of the volatility process. In particular, this allows to infer changes in the Hurst parameter of a fractional stochastic volatility process. We establish an asymptotic minimax-optimal test for this problem.

We develop change-point methods for statistics of high-frequency data. The main interest is in the stochastic volatility process of an Itô semi-martingale, the latter being discretely observed over a fixed time horizon. For a local change-point problem under high-frequency asymptotics, we construct a minimax-optimal test to discriminate continuous volatility paths from paths comprising changes. The key example is identification of volatility jumps. We prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. Moreover, we study a different global change-point problem to identify changes in the regularity of the volatility process. In particular, this allows to infer changes in the Hurst parameter of a fractional stochastic volatility process. We establish an asymptotic minimax-optimal test for this problem.

### 2017年01月26日(木)

13:00-16:00 数理科学研究科棟(駒場) 052号室

High-frequency financial data : trades and quotes databases, order flows and time resolution I, II, III

**Ioane Muni Toke 氏**(Centrale Supelec Paris)High-frequency financial data : trades and quotes databases, order flows and time resolution I, II, III

[ 講演概要 ]

I present some of the challenges associated with preparing high-frequency trades and quotes databases for statistics purposes. In a first part, I investigate TRTH tick-by-tick data on three exchanges (Paris, London and Frankfurt) and on a five-year span. I analyse the performances of a procedure of reconstruction of orders flows. This turns out to be a forensic tool assessing the quality of the database: significant technical changes affecting the exchanges are tracked through the data. Moreover, the choices made when reconstructing order flows may have consequences on the quantitative models that are calibrated afterwards on such data. I also provide a refined look at the Lee–Ready procedure and its optimal lags. Findings are in line with both financial reasoning and the analysis of an illustrative Poisson model. In a second part, I investigate Nikkei-packaged Tokyo-traded ETF data. The application the order flow reconstruction procedure underlines the differences between the TRTH and Nikkei data. In a brief last part, we will discuss the time resolution of these databases and the potential problems arising when modelling a limit order book with simple point processes.

I present some of the challenges associated with preparing high-frequency trades and quotes databases for statistics purposes. In a first part, I investigate TRTH tick-by-tick data on three exchanges (Paris, London and Frankfurt) and on a five-year span. I analyse the performances of a procedure of reconstruction of orders flows. This turns out to be a forensic tool assessing the quality of the database: significant technical changes affecting the exchanges are tracked through the data. Moreover, the choices made when reconstructing order flows may have consequences on the quantitative models that are calibrated afterwards on such data. I also provide a refined look at the Lee–Ready procedure and its optimal lags. Findings are in line with both financial reasoning and the analysis of an illustrative Poisson model. In a second part, I investigate Nikkei-packaged Tokyo-traded ETF data. The application the order flow reconstruction procedure underlines the differences between the TRTH and Nikkei data. In a brief last part, we will discuss the time resolution of these databases and the potential problems arising when modelling a limit order book with simple point processes.

### 2017年01月19日(木)

13:00-15:30 数理科学研究科棟(駒場) 052号室

Talk 1:Likelihood inference for a continuous time GARCH model

Talk 2:Nonparametric Estimation for Self-Exciting Point Processes: A Parsimonious Approach

**Feng Chen 氏**(University of New South Wales)Talk 1:Likelihood inference for a continuous time GARCH model

Talk 2:Nonparametric Estimation for Self-Exciting Point Processes: A Parsimonious Approach

[ 講演概要 ]

Talk 1:The continuous time GARCH (COGARCH) model of Kluppelberg, Lindner and Maller (2004) is a natural extension of the discrete time GARCH(1,1) model which preserves important features of the GARCH model in the discrete-time setting. For example, the COGARCH model is driven by a single source of noise as in the discrete time GARCH model, which is a Levy process in the COGARCH case, and both models can produced heavy tailed marginal returns even when the driving noise is light-tailed. However, calibrating the COGARCH model to data is a challenge, especially when observations of the COGARCH process are obtained at irregularly spaced time points. The method of moments has had some success in the case with regularly spaced data, yet it is not clear how to make it work in the more interesting case with irregularly spaced data. As a well-known method of estimation, the maximum likelihood method has not been developed for the COGARCH model, even in the quite simple case with the driving Levy process being compound Poisson, though a quasi-maximum likelihood (QML)method has been proposed. The challenge with the maximum likelihood method in this context is mainly due to the lack of a tractable form for the likelihood. In this talk, we propose a Monte Carlo method to approximate the likelihood of the compound Poisson driven COGARCH model. We evaluate the performance of the resulting maximum likelihood (ML) estimator using simulated data, and illustrate its application with high frequency exchange rate data. (Joint work with Damien Wee and William Dunsmuir).

Talk 2:There is ample evidence that in applications of self-exciting point process (SEPP) models, the intensity of background events is often far from constant. If a constant background is imposed, that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modelling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be

used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis

leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this paper we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam's razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy. Although seldom used for nonparametric function estimation in other settings, this approach is appropriate in the context of SEPP models. (Joint work with the late Peter Hall.)

Talk 1:The continuous time GARCH (COGARCH) model of Kluppelberg, Lindner and Maller (2004) is a natural extension of the discrete time GARCH(1,1) model which preserves important features of the GARCH model in the discrete-time setting. For example, the COGARCH model is driven by a single source of noise as in the discrete time GARCH model, which is a Levy process in the COGARCH case, and both models can produced heavy tailed marginal returns even when the driving noise is light-tailed. However, calibrating the COGARCH model to data is a challenge, especially when observations of the COGARCH process are obtained at irregularly spaced time points. The method of moments has had some success in the case with regularly spaced data, yet it is not clear how to make it work in the more interesting case with irregularly spaced data. As a well-known method of estimation, the maximum likelihood method has not been developed for the COGARCH model, even in the quite simple case with the driving Levy process being compound Poisson, though a quasi-maximum likelihood (QML)method has been proposed. The challenge with the maximum likelihood method in this context is mainly due to the lack of a tractable form for the likelihood. In this talk, we propose a Monte Carlo method to approximate the likelihood of the compound Poisson driven COGARCH model. We evaluate the performance of the resulting maximum likelihood (ML) estimator using simulated data, and illustrate its application with high frequency exchange rate data. (Joint work with Damien Wee and William Dunsmuir).

Talk 2:There is ample evidence that in applications of self-exciting point process (SEPP) models, the intensity of background events is often far from constant. If a constant background is imposed, that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modelling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be

used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis

leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this paper we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam's razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy. Although seldom used for nonparametric function estimation in other settings, this approach is appropriate in the context of SEPP models. (Joint work with the late Peter Hall.)

### 2017年01月16日(月)

16:50-18:00 数理科学研究科棟(駒場) 052号室

Profile likelihood approach to a large sample distribution of estimators in joint mixture model of survival and longitudinal ordered data

**広瀬勇一 氏**(University of Wellington)Profile likelihood approach to a large sample distribution of estimators in joint mixture model of survival and longitudinal ordered data

[ 講演概要 ]

We consider a semiparametric joint model that consists of item response and survival components, where these two components are linked through latent variables. We estimate the model parameters through a profile likelihood and the EM algorithm. We propose a method to derive an asymptotic variance of the estimators in this model.

We consider a semiparametric joint model that consists of item response and survival components, where these two components are linked through latent variables. We estimate the model parameters through a profile likelihood and the EM algorithm. We propose a method to derive an asymptotic variance of the estimators in this model.

### 2017年01月12日(木)

13:00-15:00 数理科学研究科棟(駒場) 052号室

yuimaGUI: a Graphical User Interface for the yuima Package

**Emanuele Guidotti 氏**(Milan University)yuimaGUI: a Graphical User Interface for the yuima Package

[ 講演概要 ]

The yuimaGUI package provides a user-friendly interface for yuima. It greatly simplifies tasks such as estimation and simulation of stochastic processes and it also includes additional tools. Some of them:

data retrieval: stock prices and economic indicators

time series clustering

change point analysis

lead-lag estimation

After a general overview of the whole interface, the yuimaGUI will be shown in real-time. All the settings and the inner workings will be discussed in detail. During this second part, you are kindly invited to ask questions whenever you feel that some problem may arise.

The yuimaGUI package provides a user-friendly interface for yuima. It greatly simplifies tasks such as estimation and simulation of stochastic processes and it also includes additional tools. Some of them:

data retrieval: stock prices and economic indicators

time series clustering

change point analysis

lead-lag estimation

After a general overview of the whole interface, the yuimaGUI will be shown in real-time. All the settings and the inner workings will be discussed in detail. During this second part, you are kindly invited to ask questions whenever you feel that some problem may arise.

### 2016年12月01日(木)

16:00-18:00 数理科学研究科棟(駒場) 052号室

On the determinant of the Malliavin matrix and density of random vector on Wiener chaos

**Ciprian Tudor 氏**(Université Lille 1)On the determinant of the Malliavin matrix and density of random vector on Wiener chaos

[ 講演概要 ]

A well-known problem in Malliavin calculus concerns the relation between the determinant of the Malliavin matrix of a random vector and the determinant of its covariance matrix. We give an explicit relation between these two determinants for couples of random vectors of multiple integrals. In particular, if the multiple integrals are of the same order, we prove that two random variables in the same Wiener chaos either admit a joint density, either are proportional and that the result is not true for random variables in Wiener chaoses of different orders.

A well-known problem in Malliavin calculus concerns the relation between the determinant of the Malliavin matrix of a random vector and the determinant of its covariance matrix. We give an explicit relation between these two determinants for couples of random vectors of multiple integrals. In particular, if the multiple integrals are of the same order, we prove that two random variables in the same Wiener chaos either admit a joint density, either are proportional and that the result is not true for random variables in Wiener chaoses of different orders.

### 2016年11月01日(火)

10:40-11:30 数理科学研究科棟(駒場) 123号室

Wavelet-based methods for high-frequency lead-lag analysis

**Yuta Koike 氏**(Tokyo Metropolitan University, JST CREST)Wavelet-based methods for high-frequency lead-lag analysis

[ 講演概要 ]

We propose a novel framework to investigate the lead-lag effect between two financial assets. Our framework bridges a gap between continuous-time modeling based on Brownian motion and the existing wavelet methods for lead-lag analysis based on discrete-time models and enables us to analyze the multi-scale structure of lead-lag effects. We also present a statistical methodology for the scale-by-scale analysis of lead-lag effects in the proposed framework and develop an asymptotic theory applicable to a situation including stochastic volatilities and irregular sampling. Finally, we report several numerical experiments to demonstrate how our framework works in practice. This talk is based on a joint work of Prof. Takaki Hayashi (Keio University).

We propose a novel framework to investigate the lead-lag effect between two financial assets. Our framework bridges a gap between continuous-time modeling based on Brownian motion and the existing wavelet methods for lead-lag analysis based on discrete-time models and enables us to analyze the multi-scale structure of lead-lag effects. We also present a statistical methodology for the scale-by-scale analysis of lead-lag effects in the proposed framework and develop an asymptotic theory applicable to a situation including stochastic volatilities and irregular sampling. Finally, we report several numerical experiments to demonstrate how our framework works in practice. This talk is based on a joint work of Prof. Takaki Hayashi (Keio University).

### 2016年11月01日(火)

11:30-12:30 数理科学研究科棟(駒場) 123号室

Second order fluctuations for zeros of arithmetic random waves

**Giovanni Peccati 氏**(University du Luxembourg)Second order fluctuations for zeros of arithmetic random waves

[ 講演概要 ]

Originally introduced by Rudnick and Wigman (2007), arithmetic random waves are Gaussian Laplace eigenfunctions on the two-dimensional torus. In this talk, I will describe the high-energy behaviour of the so-called « nodal length » (that, is the volume of the zero set) of such random objects, and show that (quite unexpectedly) it is non-central and non-universal. I will also discuss the connected problem of counting the number of intersections points of independent nodal sets (equivalent to « phase singularities » for complex waves) in the high-energy regime. Both issues are tightly connected to the arithmetic study of lattice points on circles. One key concept in our presentation is that of ‘Berry cancellation phenomenon’ (see M.V. Berry, 2002), for which an explanation in terms of chaos expansions and integration by parts (Green formula) will be provided. Based on joint works (GAFA 2016 & Preprint 2016) with D. Marinucci (Rome Tor Vergata), M. Rossi (Luxembourg) and I. Wigman (King’s College, London), and with F. Dalmao (University of Uruguay), I. Nourdin (Luxembourg) and M. Rossi (Luxembourg).

Originally introduced by Rudnick and Wigman (2007), arithmetic random waves are Gaussian Laplace eigenfunctions on the two-dimensional torus. In this talk, I will describe the high-energy behaviour of the so-called « nodal length » (that, is the volume of the zero set) of such random objects, and show that (quite unexpectedly) it is non-central and non-universal. I will also discuss the connected problem of counting the number of intersections points of independent nodal sets (equivalent to « phase singularities » for complex waves) in the high-energy regime. Both issues are tightly connected to the arithmetic study of lattice points on circles. One key concept in our presentation is that of ‘Berry cancellation phenomenon’ (see M.V. Berry, 2002), for which an explanation in terms of chaos expansions and integration by parts (Green formula) will be provided. Based on joint works (GAFA 2016 & Preprint 2016) with D. Marinucci (Rome Tor Vergata), M. Rossi (Luxembourg) and I. Wigman (King’s College, London), and with F. Dalmao (University of Uruguay), I. Nourdin (Luxembourg) and M. Rossi (Luxembourg).

### 2016年10月31日(月)

10:40-11:30 数理科学研究科棟(駒場) 123号室

Martingale expansion revisited

**Nakahiro Yoshida 氏**(University of Tokyo, Institute of Statistical Mathematics, JST CREST)Martingale expansion revisited

[ 講演概要 ]

The martingale expansion is revisited in this talk. The martingale expansion for a martingale with mixed normal limit evaluates the tangent of the quadratic variation of the martingale and the torsion of an exponential martingale under the measure transform caused by the random limit of the quadratic variation. The martingale expansion has been applied to the realized volatility, quadratic form of an Ito process, p-variation and the QLA estimators of a volatility parametric model. An interpolation in time was used in martingale expansion. We discuss relation between martingale expansion and recently developed asymptotic expansion of Skorohod integrals by interpolation of distributions (a joint work with D. Nualart).

The martingale expansion is revisited in this talk. The martingale expansion for a martingale with mixed normal limit evaluates the tangent of the quadratic variation of the martingale and the torsion of an exponential martingale under the measure transform caused by the random limit of the quadratic variation. The martingale expansion has been applied to the realized volatility, quadratic form of an Ito process, p-variation and the QLA estimators of a volatility parametric model. An interpolation in time was used in martingale expansion. We discuss relation between martingale expansion and recently developed asymptotic expansion of Skorohod integrals by interpolation of distributions (a joint work with D. Nualart).

### 2016年10月31日(月)

11:30-12:20 数理科学研究科棟(駒場) 123号室

Error analysis for approximations to one-dimensional SDEs via perturbation method

**Nobuaki Naganuma 氏**(Osaka University)Error analysis for approximations to one-dimensional SDEs via perturbation method

[ 講演概要 ]

We consider one-dimensional stochastic differential equations driven by fractional Brownian motions and adopt the Euler scheme, the Milstein type scheme and the Crank-Nicholson scheme to approximate solutions to the equations. We introduce perturbation method in order to analyze errors of the schemes. By using this method, we can express the errors in terms of directional derivatives of the solutions explicitly. We obtain asymptotic error distributions of the three schemes by combining the expression of the errors and the fourth moment theorem. This talk is based on a joint work with Prof. Shigeki Aida (Tohoku University).

We consider one-dimensional stochastic differential equations driven by fractional Brownian motions and adopt the Euler scheme, the Milstein type scheme and the Crank-Nicholson scheme to approximate solutions to the equations. We introduce perturbation method in order to analyze errors of the schemes. By using this method, we can express the errors in terms of directional derivatives of the solutions explicitly. We obtain asymptotic error distributions of the three schemes by combining the expression of the errors and the fourth moment theorem. This talk is based on a joint work with Prof. Shigeki Aida (Tohoku University).

### 2016年10月31日(月)

13:50-14:40 数理科学研究科棟(駒場) 123号室

Characterization of the convergence in total variation by Stein's method and Malliavin calculus

**Seiichiro Kusuoka 氏**(Okayama University)Characterization of the convergence in total variation by Stein's method and Malliavin calculus

[ 講演概要 ]

Recently, convergence in distributions and estimates of distances between distributions are studied by means of Stein's equation and Malliavin calculus. However, in known results, the target distributions of the convergence were some specific distributions. In this talk, we extend the target distributions to invariant probability measures of diffusion processes. Precisely speaking, we prepare Stein's equation with respect to invariant measures of diffusion processes and consider the characterization of the convergence to the invariant measure in total variation by applying Malliavin calculus. This is a joint work with Ciprian Tudor.

Recently, convergence in distributions and estimates of distances between distributions are studied by means of Stein's equation and Malliavin calculus. However, in known results, the target distributions of the convergence were some specific distributions. In this talk, we extend the target distributions to invariant probability measures of diffusion processes. Precisely speaking, we prepare Stein's equation with respect to invariant measures of diffusion processes and consider the characterization of the convergence to the invariant measure in total variation by applying Malliavin calculus. This is a joint work with Ciprian Tudor.

### 2016年10月31日(月)

14:50-15:40 数理科学研究科棟(駒場) 123号室

Parameter estimation for diffusion processes with high-frequency observations

**Teppei Ogihara 氏**(The Institute of Statistical Mathematics, JST PRESTO, JST CREST)Parameter estimation for diffusion processes with high-frequency observations

[ 講演概要 ]

We study statistical inference for security prices modeled by diffusion processes with high-frequency observations. In particular, we focus on two specific problems on analysis of high-frequency data, that is, nonsynchronous observations and the presence of observation noise called market microstructure noise. We construct a maximum-likelihood-type estimator of parameters, and study their asymptotic mixed normality. We also discuss on asymptotic efficiency of estimators.

We study statistical inference for security prices modeled by diffusion processes with high-frequency observations. In particular, we focus on two specific problems on analysis of high-frequency data, that is, nonsynchronous observations and the presence of observation noise called market microstructure noise. We construct a maximum-likelihood-type estimator of parameters, and study their asymptotic mixed normality. We also discuss on asymptotic efficiency of estimators.

### 2016年10月31日(月)

15:40-16:30 数理科学研究科棟(駒場) 123号室

Markov chain Monte Carlo for high-dimensional target distribution

**Kengo Kamatani 氏**(Osaka University, JST CREST)Markov chain Monte Carlo for high-dimensional target distribution

[ 講演概要 ]

The Markov chain Monte Carlo (MCMC) algorithms are widely used to evaluate complicated integrals in Bayesian Statistics. Since the method is not free from the curse of dimensionality, it is important to quantify the effect of the dimensionality and establish an optimal MCMC strategy in high-dimension. In this talk, I will review some high-dimensional asymptotics of MCMC initiated by Roberts et. al. 97, and explain some asymptotic properties of the MpCN algorithm. I will also mention some connection to Stein-Malliavin techniques.

The Markov chain Monte Carlo (MCMC) algorithms are widely used to evaluate complicated integrals in Bayesian Statistics. Since the method is not free from the curse of dimensionality, it is important to quantify the effect of the dimensionality and establish an optimal MCMC strategy in high-dimension. In this talk, I will review some high-dimensional asymptotics of MCMC initiated by Roberts et. al. 97, and explain some asymptotic properties of the MpCN algorithm. I will also mention some connection to Stein-Malliavin techniques.

### 2016年10月31日(月)

16:50-17:40 数理科学研究科棟(駒場) 123号室

New Functionals inequalities via Stein's discrepancies

**Giovanni Peccati 氏**(Universite du Luxembourg)New Functionals inequalities via Stein's discrepancies

[ 講演概要 ]

I will present a new set of functional inequalities involving the following four parameters associated with a given multidimensional distribution: the relative entropy, the relative Fisher information, the 2-Wasserstein distance, and the Stein discrepancy (which is a natural object arising in the framework of the Malliavin-Stein method on a Gaussian space). Our results improve the classical log-Sobolev inequality, as well Talagrand's transport inequality, and allow one to deduce new quantitative entropic limit theorems on Gaussian spaces. Joint works (JFA 2014 and GAFA 2015) with M. Ledoux (Toulouse), I. Nourdin (Luxembourg) and Y. Swan (Liège).

I will present a new set of functional inequalities involving the following four parameters associated with a given multidimensional distribution: the relative entropy, the relative Fisher information, the 2-Wasserstein distance, and the Stein discrepancy (which is a natural object arising in the framework of the Malliavin-Stein method on a Gaussian space). Our results improve the classical log-Sobolev inequality, as well Talagrand's transport inequality, and allow one to deduce new quantitative entropic limit theorems on Gaussian spaces. Joint works (JFA 2014 and GAFA 2015) with M. Ledoux (Toulouse), I. Nourdin (Luxembourg) and Y. Swan (Liège).

### 2016年10月31日(月)

17:40-18:30 数理科学研究科棟(駒場) 123号室

Stochastic geometry and Malliavin calculus on configuration spaces

**Giovanni Peccati 氏**(Université du Luxembourg)Stochastic geometry and Malliavin calculus on configuration spaces

[ 講演概要 ]

I will present some recent advances in the domain of quantitative limit theorems for geometric Poisson functionals, associated e.g. with random geometric graphs and random tessellations, obtained by means of Malliavin calculus techniques. One of our main results consists in a general (optimal) Berry-Esseen bound for stabilizing functionals, based on Stein’s method, iterated Poincaré inequalities and a variant of Mehler’s formula. Based on several joint works with S. Bourguin, R. Lachièze-Rey, G. Last and M. Schulte, as well as on the recent monograph that I co-edited with M. Reitzner.

I will present some recent advances in the domain of quantitative limit theorems for geometric Poisson functionals, associated e.g. with random geometric graphs and random tessellations, obtained by means of Malliavin calculus techniques. One of our main results consists in a general (optimal) Berry-Esseen bound for stabilizing functionals, based on Stein’s method, iterated Poincaré inequalities and a variant of Mehler’s formula. Based on several joint works with S. Bourguin, R. Lachièze-Rey, G. Last and M. Schulte, as well as on the recent monograph that I co-edited with M. Reitzner.

### 2016年10月11日(火)

16:50-18:00 数理科学研究科棟(駒場) 052号室

ネットワーク理論を応用した相関クラスタリングによる株価の自動分類

**磯貝 孝 氏**(首都大学東京)ネットワーク理論を応用した相関クラスタリングによる株価の自動分類

[ 講演概要 ]

本研究では、東証１部上場の株価の日次収益率データを用いて、株価の銘柄間の

相関行列を推定し、銘柄をノードとする相関ネットワークを構築した。この相関

ネットワークに対して複雑ネットワーク理論で用いられているクラスタリング手

法を適用し、業種分類とは別の銘柄グループを構成し、それらの銘柄グループが

どのような特徴を持つのか、業種分類とどのような関係があるのか、などについ

て明らかとなった点について紹介する。

本研究では、東証１部上場の株価の日次収益率データを用いて、株価の銘柄間の

相関行列を推定し、銘柄をノードとする相関ネットワークを構築した。この相関

ネットワークに対して複雑ネットワーク理論で用いられているクラスタリング手

法を適用し、業種分類とは別の銘柄グループを構成し、それらの銘柄グループが

どのような特徴を持つのか、業種分類とどのような関係があるのか、などについ

て明らかとなった点について紹介する。

### 2016年08月10日(水)

13:00-14:30 数理科学研究科棟(駒場) 117号室

Malliavin calculus and normal approximations

**David Nualart 氏**(Kansas University)Malliavin calculus and normal approximations

### 2016年08月09日(火)

13:00-16:30 数理科学研究科棟(駒場) 117号室

本ワークショップ・レクチャーは統計数学セミナー共催であり，JST CRESTによってサポートされています.

Malliavin calculus and normal approximations

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=180

本ワークショップ・レクチャーは統計数学セミナー共催であり，JST CRESTによってサポートされています.

**David Nualart 氏**(Kansas University)Malliavin calculus and normal approximations

[ 講演概要 ]

The purpose of these lectures is to introduce some recent results on the application of Malliavin calculus combined with Stein's method to normal approximation. The Malliavin calculus is a differential calculus on the Wiener space. First, we will present some elements of Malliavin calculus, defining the basic differential operators: the derivative, its adjoint called the divergence operator and the generator of the Ornstein-Uhlenbeck semigroup. The behavior of these operators on the Wiener chaos expansion will be discussed. Then, we will introduce the Stein's method for normal approximation, which leads to general bounds for the Kolmogorov and total variation distances between the law of a Brownian functional and the standard normal distribution. In this context, the integration by parts formula of Malliavin calculus will allow us to express these bounds in terms of the Malliavin operators. We will present the application of this methodology to derive the Fourth Moment Theorem for a sequence of multiple stochastic integrals, and we will discuss some results on the uniform convergence of densities obtained using Malliavin calculus techniques. Finally, examples of functionals of Gaussian processes, such as the fractional Brownian motion, will be discussed.

[ 参考URL ]The purpose of these lectures is to introduce some recent results on the application of Malliavin calculus combined with Stein's method to normal approximation. The Malliavin calculus is a differential calculus on the Wiener space. First, we will present some elements of Malliavin calculus, defining the basic differential operators: the derivative, its adjoint called the divergence operator and the generator of the Ornstein-Uhlenbeck semigroup. The behavior of these operators on the Wiener chaos expansion will be discussed. Then, we will introduce the Stein's method for normal approximation, which leads to general bounds for the Kolmogorov and total variation distances between the law of a Brownian functional and the standard normal distribution. In this context, the integration by parts formula of Malliavin calculus will allow us to express these bounds in terms of the Malliavin operators. We will present the application of this methodology to derive the Fourth Moment Theorem for a sequence of multiple stochastic integrals, and we will discuss some results on the uniform convergence of densities obtained using Malliavin calculus techniques. Finally, examples of functionals of Gaussian processes, such as the fractional Brownian motion, will be discussed.

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=180

### 2016年08月06日(土)

10:00-17:10 数理科学研究科棟(駒場) 123号室

本ワークショップ・レクチャーは統計数学セミナー共催であり，JST CRESTによってサポートされています．

Asymptotic expansion of variations

LAMN property and optimal estimation for diffusion with non synchronous observations

Approximation schemes for stochastic differential equations driven by a fractional Brownian motion

Parameter estimation for fractional Ornstein-Uhlenbeck processes

Stein's equations for invariant measures of diffusions processes and their applications via Malliavin calculus

Asymptotic expansion of a nonlinear oscillator with a jump diffusion

[ 参考URL ]

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=179

本ワークショップ・レクチャーは統計数学セミナー共催であり，JST CRESTによってサポートされています．

**Nakahiro Yoshida 氏**(University of Tokyo, Institute of Statistical Mathematics, and JST CREST) 10:00-10:50Asymptotic expansion of variations

**Teppei Ogihara 氏**(The Institute of Statistical Mathematics, JST PRESTO, and JST CREST) 11:00-11:50LAMN property and optimal estimation for diffusion with non synchronous observations

**David Nualart 氏**(Kansas University) 13:10-14:00Approximation schemes for stochastic differential equations driven by a fractional Brownian motion

**David Nualart 氏**(Kansas University) 14:10-15:00Parameter estimation for fractional Ornstein-Uhlenbeck processes

**Seiichiro Kusuoka 氏**(Okayama University) 15:20-16:10Stein's equations for invariant measures of diffusions processes and their applications via Malliavin calculus

**Yasushi Ishikawa 氏**(Ehime University) 16:20-17:10Asymptotic expansion of a nonlinear oscillator with a jump diffusion

[ 参考URL ]

http://www2.ms.u-tokyo.ac.jp/probstat/?page_id=179

### 2016年07月26日(火)

13:00-14:30 数理科学研究科棟(駒場) 052号室

本講演は大阪大学で行い，東京大学へWeb配信いたします.

Multilevel Particle Filters

本講演は大阪大学で行い，東京大学へWeb配信いたします.

**Ajay Jasra 氏**(National University of Singapore)Multilevel Particle Filters

[ 講演概要 ]

In this talk the filtering of partially observed diffusions,

with discrete-time observations, is considered.

It is assumed that only biased approximations of the diffusion can be

obtained, for choice of an accuracy parameter indexed by $l$.

A multilevel estimator is proposed, consisting of a telescopic sum of

increment estimators associated to the successive levels.

The work associated to $\cO(\varepsilon^2)$ mean-square error between

the multilevel estimator and average with respect to the filtering

distribution is shown to scale optimally, for example as

$\cO(\varepsilon^{-2})$ for optimal rates of convergence of the

underlying diffusion approximation.

The method is illustrated on several examples.

In this talk the filtering of partially observed diffusions,

with discrete-time observations, is considered.

It is assumed that only biased approximations of the diffusion can be

obtained, for choice of an accuracy parameter indexed by $l$.

A multilevel estimator is proposed, consisting of a telescopic sum of

increment estimators associated to the successive levels.

The work associated to $\cO(\varepsilon^2)$ mean-square error between

the multilevel estimator and average with respect to the filtering

distribution is shown to scale optimally, for example as

$\cO(\varepsilon^{-2})$ for optimal rates of convergence of the

underlying diffusion approximation.

The method is illustrated on several examples.

### 2016年06月21日(火)

13:00-15:00 数理科学研究科棟(駒場) 052号室

New Classes and Methods in YUIMA package

**Lorenzo Mercuri 氏**(University of Milan)New Classes and Methods in YUIMA package

[ 講演概要 ]

In this talk, we present three new classes recently introduced in YUIMA package.

These classes allow the user to manage three different problems:

・Construction of a multidimensional stochastic differential equation driven by a general multivariate Levy process. In particular we show how to define and then simulate a SDE driven by a multivariate Variance Gamma process.

・Definition and simulation of a functional of a general SDE.

・Definition and simulation of the integral of an object from the class yuima.model. In particular, we are able to evaluate Riemann Stieltjes integrals,deterministic integrals with random integrand and stochastic integrals.

Numerical examples are given in order to explain the new methods and classes.

In this talk, we present three new classes recently introduced in YUIMA package.

These classes allow the user to manage three different problems:

・Construction of a multidimensional stochastic differential equation driven by a general multivariate Levy process. In particular we show how to define and then simulate a SDE driven by a multivariate Variance Gamma process.

・Definition and simulation of a functional of a general SDE.

・Definition and simulation of the integral of an object from the class yuima.model. In particular, we are able to evaluate Riemann Stieltjes integrals,deterministic integrals with random integrand and stochastic integrals.

Numerical examples are given in order to explain the new methods and classes.