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Market Microstructure: Confronting Many Viewpoints by Thierry Foucault, Mathieu Rosenbaum, Charles-Albert Lehalle, Frédéric Abergel, Jean-Philippe Bouchaud

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4

Statistical Inference for Volatility and Related Limit Theorems

Nakahiro Yoshida

4.1 INTRODUCTION

This chapter provides a brief overview of some recent developments in statistical inference for stochastic processes from probabilistic and statistical aspects. We will discuss mainly three topics, that is, quasi likelihood analysis for semimartingales, nonsynchronous covariance estimation, and asymptotic expansion that can apply to finance.

In Sections 4.2 and 4.3, we will discuss quasi likelihood analysis (QLA) for semimartingales. In this article, the QLA means an asymptotic theory that provides asymptotic distribution of the quasi maximum likelihood and quasi Bayesian estimators (polynomial type), large deviation estimates for the quasi likelihood random field, and tail probability estimates for these estimators with their convergence of moments as a result. One cannot avoid such kinds of tail probability estimates when developing the very basic fields in standard theoretical statistics.

Section 4.4 is devoted to the problem of nonsynchronous covariance estimation. Statistical inference for stochastic processes under an irregular sampling scheme requires new developments in limit theorems. Among various irregularities, the nonsynchronicity gives challenging problems. Ad hoc interpolation causes bias in covariance estimation. The problem is how to construct a suitable estimator and to prove probabilistic performance of the estimator. Nonsynchronicity is not standard in stochastic ...

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