Page 103 - Contributed Paper Session (CPS) - Volume 2

CPS1443 Shogo H.N. et al.

Adaptive estimators for noisily observed
diffusion processes
1,2
1
Shogo H Nakakita ; Masayuki Uchida
1 Graduate School of Engineering Science, Osaka University
2 Center for Mathematical Modeling and Data Science, Osaka University and CREST JST

Abstract
We propose adaptive quasi-likelihood functions and adaptive maximum-
likelihood-type estimators for discretely and noisily observed ergodic diffusion
processes, and show the consistency, asymptotic normality and convergence
of moments of the estimators. We also demonstrate computational simulation
study with the proposed method and compare the result with that of an
existent method which does not concern noise existence.

Keywords
Diffusion processes; high-frequency data; observation noise; quasi-likelihood
analysis

1. Introduction

Let us define the -dimensional ergodic diffusion process { } defined
≥0
by the following stochastic differential equation (SDE):

d = ( , )d + ( , )d , = ,

0
0

where { } is an -dimensional Wiener process, is a -dimensional
≥0
0
random variable independent of { } , ∈ Θ ⊂ 1 and ∈ Θ ⊂ 2 are
1
≥0
2
unknown parameters, Θ is bounded, open and convex sets in admitting

Sobolev’s inequalities (see Adams and Fournier, 2003; Yoshida, 2011) for =
⋆

⋆
⋆

1, 2, = ( , ) is the true value of the parameter, and : × Θ → ⊗
1
and : × Θ → are known functions.

2
Our purpose is to model some phenomena observed at high frequency
such that stock prices, wind velocity and EEG with the parametric diffusion
process { } . Let us denote the discretisation step of observation time as
≥0
ℎ > 0 and statistical inference for discretely observed diffusion processes

{ } has been enthusiastically researched for the last few decades (e.g.,
ℎ =0,…,
see Bibby and Sørensen, 1995; Florens-Zmirou, 1989; Kessler, 1995, 1997,
Uchida and Yoshida, 2012, 2014; Yoshida, 1992, 2011). Although these
researches are about inference based on observation of { }
ℎ =0,…,
indicating that we can obtain correct values of { } on the time mesh
ℎ =0,…,
{ℎ } , sometimes there exist exogenous noises contaminating our
=0,…,
observation for the phenomena of interest known as microstructure noise in
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