# Statistical inference for stochastic ordinary and partial differential equations

### Conference

Category: Bernoulli Society for Mathematical Statistics and Probability (BS)

### Abstract

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.