Professor Nakahiro Yoshida will present new developments in the quasi-likelihood analysis for stochastic processes. The quasi-likelihood analysis (QLA) is a systematic framework for statistical inference for stochastic processes. Based on the polynomial type large deviation inequality, it ensures asymptotic properties of the Bayesian estimator for nonlinear stochastic processes, for example. Recently, QLA has found new applications to various large-scale spatio-temporal dependence models such as degenerate diffusion processes and point processes for limit order book, besides, the theory has been extended to sparse estimation, and inference for non-identifiable models. We will discuss recent developments of the theory of QLA and its applications.
Professor Arnaud Gloter will talk about locally differentially private drift parameter estimation for iid paths of diffusion processes. The problem of drift parameter estimation is addressed for N discretely observed iid SDEs, considering the additional constraints that only privatized data can be published and used for inference. The concept of local differential privacy is formally introduced for a system of stochastic differential equations. The aim is to estimate the drift parameter by proposing a contrast function based on a pseudo-likelihood approach. A suitably scaled Laplace noise is incorporated to satisfy the privacy requirement. Our main results consist of deriving explicit conditions on the privacy level for which the associated estimator is proven to be consistent. This holds true as the discretization step approaches zero and the number of processes N tends to infinity.
Professor Ciprian Tudor will present inference for the nonlinear stochastic heat equation. We consider the nonlinear stochastic heat equation with fractional Laplacian, driven by the Gaussian space-time white noise and we analyse the asymptotic behavior of the quadratic and higher order variations of its mild solution. The idea is to approximate the increments of the solution to the nonlinear heat equation with those of the solution to the linear heat equation (which is related to the fractional Brownian motion). Based on these variations, we construct estimators for several parameters that may appear in such a model: the drift and diffusion parameters or the parameter associated with the order of the fractional Laplacian.
Professor Masayuki Uchida will talk about estimation for a discretely observed stochastic partial differential equation (SPDE) in two space dimensions based on temporal and spatial increments. Minimum contrast estimators (MCEs) of the unknown parameters in the SPDE are obtained by utilizing high-frequency spatio-temporal data. An approximate coordinate process is constructed from the MCEs and the spatio-temporal data. In addition, an adaptive estimator of the remaining unknown parameter in the SPDE is derived from the temporally thinned data obtained from the approximate coordinate process. The consistency and asymptotic normality of the adaptive estimator are established.