The development of information and measurement technology has made it possible to obtain high-frequency time series data and high-frequency spatio-temporal data. It allows us to study statistical modeling for both stochastic differential equations from high-frequency data and stochastic partial differential equations from high-frequency spatio-temporal data. With the above background, there is an urgent need to develop statistical inference for stochastic ordinary and partial differential equations. Statistical inference of stochastic differential equations and stochastic partial differential equations plays an important role in the analysis of time-dependent phenomena. Diffusion processes, Lévy driven stochastic processes defined by stochastic differential equations, and the stochastic heat equation and the sea surface temperature model defined by the stochastic partial differential equation are important classes of continuous time stochastic processes. In this session, four speakers are invited to give a talk about statistical inference for continuous time stochastic processes based on discrete observations. They are Professors Michael Sorensen from University of Copenhagen, Nakahiro Yoshida from Kyushu University of Tokyo, Markus Bibinger from University of Würzburg and Masayuki Uchida from Osaka University. Professor Michael Sorensen is an expert in statistical inference for stochastic processes including continuous time stochastic processes such as models given by stochastic differential equations and jump processes. In particular, he is a pioneer in statistics based on martingale estimating functions for discretely observed stochastic differential equations. Professor Nakahiro Yoshida is an authority in the fields of quasi-likelihood analysis for stochastic processes including stochastic differential equations, and asymptotic expansion of statistics for stochastic differential equations in both ergodic and non-ergodic cases. Furthermore, he proved the mathematical validity of asymptotic expansion of Skorohod integrals converging to a mixed normal limit. Professor Markus Bibinger has made significant achievements in covariance estimation for stochastic differential equations based on noisy and nonsynchronous high‐frequency data. Recently he obtained minimum contrast estimators of stochastic partial differential equations in one space dimensions based on high-frequency spatio-temporal data. Professor Masayuki Uchida is a researcher of statistical inference of discretely observed stochastic ordinary and partial differential equations. Specifically, he has studied the problem of estimating unknown parameters for stochastic partial differential equations in two space dimensions from high-frequency spatio-temporal data.

Professor Michael Sørensen will talk about simulating diffusion bridges using confluent diffusions. Simulation-based likelihood inference for discretely observed stochastic differential equation models requires simulation techniques for diffusions conditioned on hitting a given endpoint, so-called diffusion bridges. The talk presents a Markov chain Monte Carlo algorithm that combines two previous methods to obtain an algorithm which 1) is exact in the sense that there is no discretization error, 2) has computational cost that is linear in the duration of the bridge, and 3) provides bounds on local maxima and minima of the simulated trajectory. The method is similar to that proposed by Bladt and Sørensen in 2014, but here the unconditional sample paths are simulated by a method without discretization error.
Professor Nakahiro Yoshida will present recent developments in statistics with the Malliavin calculus. Around 1990, the Malliavin calculus was first applied by the author to higher order inference for stochastic differential equations, and it induced various applications in statistics and finance. In 90’s, the higher order asymptotic distribution theory was extended by the martingale expansion and the expansion for continuous-time Markov processes, both based on the Malliavin calculus. The martingale expansion was extended around 2010 to non-ergodic statistics covering volatility estimation with high frequency data. Recently, essential generalizations of the theory have taken place toward general Wiener functionals, and Skorohod integrals in non-ergodic statistics. By these new engines, it has been possible to approach anticipative functionals, functionals of a fractional Brownian motion like fractional diffusions, and related statistical problems. In this talk, some of recent developments on the front of the theories and their applications in statistics are introduced.
Professor Markus Bibinger will talk about efficient parameter estimation for parabolic SPDEs based on a log-linear model. We present statistical methods to calibrate dynamic models based on stochastic partial differential equations (SPDEs). We construct estimators for the parameters of parabolic SPDEs based on discrete observations in time and space of a solution on a bounded domain. We point out the relation of power variation statistics to the response variable of a log-linear model. This allows to conclude about efficiency and minimal variances. We establish central limit theorems under high-frequency asymptotics. The asymptotic variances are smaller compared to existing estimation methods and asymptotic confidence intervals are directly feasible. We demonstrate the efficiency gains numerically and in Monte Carlo simulations.
Professor Masayuki Uchida will present parametric estimation of a parabolic linear second order stochastic partial differential equation (SPDE) with two-dimensional space based on high-frequency spatio-temporal data. The minimum contrast estimator (MCE) of the diffusivity parameter and the curvature parameter in the SPDE are obtained by using the thinned data in space. The approximate coordinate process is derived from the MCE and the discrete observations. The adaptive estimator of the rest of coefficient parameters in the SPDE is constructed by using the thinned data in time obtained from the approximate coordinate process. It is shown that the adaptive estimator has consistency and asymptotic normality.

Organiser: Prof. Masayuki Uchida 

Chair: Hiroki Masuda

Speaker: Prof. Michael Sørensen 

Speaker: Prof. Masayuki Uchida 

Speaker: Prof. Dr. Nakahiro Yoshida 

Speaker: Markus Bibinger