概要Overview

 In this talk, we will revisit some results from the monograph “Statistics of Random Processes, Vol. II, Springer, 2001” by R.Liptser & A.Shiryaev. For diffusion-type processes, we will discuss the sequential drift parameter estimation (including a fixed sample size case). We will present the non-asymptotic formulae for moments of the maximum likelihood estimates (MLE) λ_{τ}, where τ is a stopping time.  Then we will consider the mean-reverting ergodic diffusion process X={X_{t},0≤t≤τ} with a diffusion coefficient of the form σ_{t}(X)=X_{t}^{γ}, γ≥0. In particular, we will present the analytical results for exact values of bias and mean-square-error (MSE) of MLE λ_{T} when T=const (i.e. a fixed sample size) and for λ_{τ_{H}} with the stopping time τ_{H} that guarantees the predefined variance of (1/H). For comparative analysis, we will also provide some analytical and numerical results.

This presentation is based on recent papers jointly with Dr. Nino Kordzakhia (Macquarie University, Sydney) and Albert Shiryaev (Steklov Mathematical Institute).

講師ご略歴Profile for speakers

Alexander Novikov is the Emeritus Professor at the Department of Mathematical Sciences, University of Technology Sydney.

Prior to this, since 1970 he was a Leading Research Fellow at the Steklov Mathematical Institute. Alexander lives in Australia since 1996. His research interest includes stochastic processes, statistics of random processes, sequential analysis, random fields, and mathematical finance.

He received a Ph.D. in Mathematics in 1972 and his Doctor of Science degree in 1982, both from the Steklov Mathematical Institute.

Alexander is the author of Novikov’s Condition (see https://en.wikipedia.org/wiki/Novikov%27s_condition).

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