## 統計数学セミナー

過去の記録 ～11/14｜次回の予定｜今後の予定 11/15～

開催情報 | 火曜日 13:00～14:10 数理科学研究科棟(駒場) 052号室 |
---|---|

担当者 | http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/ |

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**

### 2018年10月30日(火)

15:30-16:40 数理科学研究科棟(駒場) 126号室

Asymptotic expansion for random vectors

**Ciprian A. Tudor 氏**(Université de Lille 1, Université de Panthéon-Sorbonne Paris 1)Asymptotic expansion for random vectors

[ 講演概要 ]

We develop the asymptotic expansion theory for vector-valued sequences $F_{N}$ of random variables. We find the second-order term in the expansion of the density of $F_{N}$, based on assumptions in terms of the convergence of the Stein-Malliavin matrix associated to the sequence $F_{N}$ . Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of $F_{N}$ and we discuss the main ideas on higher order asymptotic expansion. We illustrate our results by several examples.

We develop the asymptotic expansion theory for vector-valued sequences $F_{N}$ of random variables. We find the second-order term in the expansion of the density of $F_{N}$, based on assumptions in terms of the convergence of the Stein-Malliavin matrix associated to the sequence $F_{N}$ . Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the asymptotic expansion of the density of $F_{N}$ and we discuss the main ideas on higher order asymptotic expansion. We illustrate our results by several examples.

### 2018年05月23日(水)

14:00-15:10 数理科学研究科棟(駒場) 052号室

"yuima.law": From mathematical representation of general Lévy processes to a numerical implementation

**Lorenzo Mercuri 氏**(University of Milan)"yuima.law": From mathematical representation of general Lévy processes to a numerical implementation

[ 講演概要 ]

We present a new class called yuima.law that refers to the mathematical description of a general Lévy process used in the formal definition of a general Stochastic Differential Equation. The final aim is to have an object, defined by the user, that contains all possible information about the Lévy process considered. This class creates a link between YUIMA and other R packages available on CRAN that manage specific Lévy processes.

An example of yuima.law is shown based the Mixed Tempered Stable(MixedTS) Lévy processes. A review of the univariate MixedTS is given and some new results on the asymptotic tail behaviour are derived. The multivariate version of the Mixed Tempered Stable, which is a generalisation of the Normal Variance Mean Mixtures, is discussed. Characteristics of this distribution, its capacity in fitting tails and in capturing dependence structure between components are investigated.

We present a new class called yuima.law that refers to the mathematical description of a general Lévy process used in the formal definition of a general Stochastic Differential Equation. The final aim is to have an object, defined by the user, that contains all possible information about the Lévy process considered. This class creates a link between YUIMA and other R packages available on CRAN that manage specific Lévy processes.

An example of yuima.law is shown based the Mixed Tempered Stable(MixedTS) Lévy processes. A review of the univariate MixedTS is given and some new results on the asymptotic tail behaviour are derived. The multivariate version of the Mixed Tempered Stable, which is a generalisation of the Normal Variance Mean Mixtures, is discussed. Characteristics of this distribution, its capacity in fitting tails and in capturing dependence structure between components are investigated.

### 2018年05月23日(水)

15:30-16:40 数理科学研究科棟(駒場) 052号室

Latest Development in yuimaGUI - Interactive Platform for Computational Statistics and Finance

**Emanuele Guidotti 氏**(University of Milan)Latest Development in yuimaGUI - Interactive Platform for Computational Statistics and Finance

[ 講演概要 ]

The yuimaGUI package provides a user-friendly interface for the yuima package, including additional tools related to Quantitative Finance. It greatly simplifies tasks such as estimation and simulation of stochastic processes, data retrieval, time series clustering, change point and lead-lag analysis. Today we are going to discuss the latest development in yuimaGUI, extending the Platform with multivariate modeling and simulation, Levy processes, Point processes, broader model selection tools and more general distributions thanks to the new yuima-Law object.

The yuimaGUI package provides a user-friendly interface for the yuima package, including additional tools related to Quantitative Finance. It greatly simplifies tasks such as estimation and simulation of stochastic processes, data retrieval, time series clustering, change point and lead-lag analysis. Today we are going to discuss the latest development in yuimaGUI, extending the Platform with multivariate modeling and simulation, Levy processes, Point processes, broader model selection tools and more general distributions thanks to the new yuima-Law object.

### 2018年05月08日(火)

15:00-16:10 数理科学研究科棟(駒場) 052号室

LAN property for stochastic differential equations driven by fractional Brownian motion of Hurst parameter 1/4 < H < 1/2

**千葉 航平 氏**(東京大学)LAN property for stochastic differential equations driven by fractional Brownian motion of Hurst parameter 1/4 < H < 1/2

[ 講演概要 ]

We consider the problem of estimating the drift parameter of solution to the stochastic differential equation driven by a fractional Brownian motion with Hurst parameter less than 1/2 under complete observation. We derive a formula for the likelihood ratio and prove local asymptotic normality when 1/4 < H < 1/2. Our result shows that the convergence rate is $T^{-1/2}$ for the parameters satisfying a certain equation and $T^{-(1-H)}$ for the others.

In this talk, we outline the proof of local asymptotic normality and explain how the different rates of convergence occur and where we use the assumption H > 1/4. We also mention some remaining problems and future directions. This talk is based on arXiv:1804.04108.

We consider the problem of estimating the drift parameter of solution to the stochastic differential equation driven by a fractional Brownian motion with Hurst parameter less than 1/2 under complete observation. We derive a formula for the likelihood ratio and prove local asymptotic normality when 1/4 < H < 1/2. Our result shows that the convergence rate is $T^{-1/2}$ for the parameters satisfying a certain equation and $T^{-(1-H)}$ for the others.

In this talk, we outline the proof of local asymptotic normality and explain how the different rates of convergence occur and where we use the assumption H > 1/4. We also mention some remaining problems and future directions. This talk is based on arXiv:1804.04108.

### 2018年03月15日(木)

16:00-17:10 数理科学研究科棟(駒場) 052号室

On Hypotheses testing for discretely observed SDE (Joint work with Alessandro De Gregorio, University of Rome)

**Stefano Iacus 氏**(University of Milan)On Hypotheses testing for discretely observed SDE (Joint work with Alessandro De Gregorio, University of Rome)

[ 講演概要 ]

In this talk we consider parametric hypotheses testing for discretely observed ergodic diffusion processes. We present the different test statistics proposed in literature and recall their asymptotic properties. We also compare the empirical performance of different tests in the case of small sample sizes.

In this talk we consider parametric hypotheses testing for discretely observed ergodic diffusion processes. We present the different test statistics proposed in literature and recall their asymptotic properties. We also compare the empirical performance of different tests in the case of small sample sizes.

### 2018年03月02日(金)

15:00-16:10 数理科学研究科棟(駒場) 270号室

"Estimating functions for SDE driven by stable Lévy processes"

Joint work with Emmanuelle Clément (Ecole Centrale)

**Arnaud Gloter 氏**(Université d'Evry Val d'Essonne)"Estimating functions for SDE driven by stable Lévy processes"

Joint work with Emmanuelle Clément (Ecole Centrale)

[ 講演概要 ]

In this talk we will discuss about parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an α-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of α ∈ (0, 2).

In this talk we will discuss about parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an α-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of α ∈ (0, 2).

### 2018年02月04日(日)

12:00-18:00 数理科学研究科棟(駒場) 118号室

Nonlinear Economic Time Series Models

Analytic, representation and statistical aspects related to fractional Gaussian processes.

**Yukai Yang 氏**(Uppsala University) 12:30-15:00Nonlinear Economic Time Series Models

[ 講演概要 ]

The lecture goes through several chapters in the book “Modelling Nonlinear Economic Time Series” by Teräsvirta, Tjøstheim and Granger in 2010. The lecture serves as an introduction for the students and researchers who are interested in this area. It introduces a number of examples of families of nonlinear time series parametric models in economic theory. It also talks about testing linearity against parametric alternatives with the presence of a characterization of the identification problem in many situations. Different ways of solving the identification problem are presented and their merits and disadvantages are discussed.

The lecture goes through several chapters in the book “Modelling Nonlinear Economic Time Series” by Teräsvirta, Tjøstheim and Granger in 2010. The lecture serves as an introduction for the students and researchers who are interested in this area. It introduces a number of examples of families of nonlinear time series parametric models in economic theory. It also talks about testing linearity against parametric alternatives with the presence of a characterization of the identification problem in many situations. Different ways of solving the identification problem are presented and their merits and disadvantages are discussed.

**Yuliia Mishura 氏**(The Taras Shevchenko National University of Kiev ) 15:30-18:00Analytic, representation and statistical aspects related to fractional Gaussian processes.

[ 講演概要 ]

We consider the properties of fractional Gaussian processes whose covariance function is situated between two self-similarities, or, in other words, these processes belong to the generalized quasi-helix, according to geometric terminology of Kahane. For such processes we consider the two-sided bounds for maximal functionals and the representation results. We consider stochastic differential equations involving fractional Brownian motion and present also several results on statistical estimations for them.

We consider the properties of fractional Gaussian processes whose covariance function is situated between two self-similarities, or, in other words, these processes belong to the generalized quasi-helix, according to geometric terminology of Kahane. For such processes we consider the two-sided bounds for maximal functionals and the representation results. We consider stochastic differential equations involving fractional Brownian motion and present also several results on statistical estimations for them.

### 2018年02月02日(金)

13:30-14:40 数理科学研究科棟(駒場) 052号室

Estimation of ratios of intensities in a Cox-type model of limit order books

**Ioane Muni Toke 氏**(Centrale Supelec Paris)Estimation of ratios of intensities in a Cox-type model of limit order books

[ 講演概要 ]

We introduce a Cox-type model for relative intensities of orders flows in a limit order book. The Cox-like intensities of the counting processes of events are assumed to share an unobserved and unspecified baseline intensity, which in finance can be identified to a global market activity affecting all events. The model is formulated in terms of relative responses of the intensities to covariates, and relative parameters can be estimated by quasi likelihood maximization. Consistency and asymptotic normality of the estimators are proven. Computationally intensive inferences are run on large samples of tick-by-tick data (35+ stocks and 220+ trading days, adding to more than one billion events). Penalization methods are also investigated. Results of the model are interpreted in terms of probability of occurrence of events. Excellent agreement with empirical data is found. Estimated model reproduces known empirical facts on imbalance, spread and queue sizes, and helps identifying trading signals of interests on a given stock.

Joint work with N.Yoshida.

We introduce a Cox-type model for relative intensities of orders flows in a limit order book. The Cox-like intensities of the counting processes of events are assumed to share an unobserved and unspecified baseline intensity, which in finance can be identified to a global market activity affecting all events. The model is formulated in terms of relative responses of the intensities to covariates, and relative parameters can be estimated by quasi likelihood maximization. Consistency and asymptotic normality of the estimators are proven. Computationally intensive inferences are run on large samples of tick-by-tick data (35+ stocks and 220+ trading days, adding to more than one billion events). Penalization methods are also investigated. Results of the model are interpreted in terms of probability of occurrence of events. Excellent agreement with empirical data is found. Estimated model reproduces known empirical facts on imbalance, spread and queue sizes, and helps identifying trading signals of interests on a given stock.

Joint work with N.Yoshida.

### 2017年11月16日(木)

13：00-16：00 数理科学研究科棟(駒場) 123号室

スパース推定法における自由度について

**二宮嘉行 氏**(九州大学)スパース推定法における自由度について

[ 講演概要 ]

スパース推定では、どれだけスパースに推定するかを決める、つまり罰則項の強弱を決めるチューニングパラメータの値を選択する必要がある。そして、その選択のための情報量規準で用いられる自由度が、主に正規線形回帰モデルの設定のもとで計算されている。本発表では、その自由度について紹介した後、それを形式上一般化線形モデルなどに拡張する情報量規準を導く。具体的には、古典的な統計的漸近理論に基づき、AIC型の情報量規準を導出する。

スパース推定では、どれだけスパースに推定するかを決める、つまり罰則項の強弱を決めるチューニングパラメータの値を選択する必要がある。そして、その選択のための情報量規準で用いられる自由度が、主に正規線形回帰モデルの設定のもとで計算されている。本発表では、その自由度について紹介した後、それを形式上一般化線形モデルなどに拡張する情報量規準を導く。具体的には、古典的な統計的漸近理論に基づき、AIC型の情報量規準を導出する。

### 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.