Statistical Inference For Fractional Diffusion Processes

Stochastic processes are widely used for model building in the social, physical, engineering and life sciences as well as in financial economics. In model building, statistical inference for stochastic processes is of great importance from both a theoretical and an applications point of view.


This book deals with Fractional Diffusion Processes and statistical inference for such stochastic processes. The main focus of the book is to consider parametric and nonparametric inference problems for fractional diffusion processes when a complete path of the process over a finite interval is observable.


Key features:


* Introduces self-similar processes, fractional Brownian motion and stochastic integration with respect to fractional Brownian motion.

* Provides a comprehensive review of statistical inference for processes driven by fractional Brownian motion for modelling long range dependence.

* Presents a study of parametric and nonparametric inference problems for the fractional diffusion process.

* Discusses the fractional Brownian sheet and infinite dimensional fractional Brownian motion.



* Includes recent results and developments in the area of statistical inference of fractional diffusion processes.






Researchers and students working on the statistics of fractional diffusion processes and applied mathematicians and statisticians involved in stochastic process modelling will benefit from this book.

B.L.S. Prakasa Rao, Department of Mathematics and Statistics, University of Hyderabad, India. Professor Rao is one of the world's foremost researchers in this complex area of probability theory.

Preface





1 Fractional Brownian Motion and Related Processes


1.1 Introduction


1.2 Self-similar processes


1.3 Fractional Brownian motion


1.4 Stochastic differential equations driven by fBm


1.5 Fractional Ornstein-Uhlenbeck type process


1.6 Mixed fractional Brownian motion


1.7 Donsker type approximation for fBm with Hurst index H > 12


1.8 Simulation of fractional Brownian motion


1.9 Remarks on application of modelling by fBm in mathematical finance


1.10 Path wise integration with respect to fBm








2 Parametric Estimation for Fractional Diffusion Processes


2.1 Introduction


2.2 Stochastic differential equations and local asymptotic normality


2.3 Parameter estimation for linear SDE


2.4 Maximum likelihood estimation


2.5 Bayes estimation


2.6 Berry-Esseen type bound for MLE


2.7 _-upper and lower functions for MLE


2.8 Instrumental variable estimation








3 Parametric Estimation for Fractional Ornstein-Uhlenbeck Type Process


3.1 Introduction


3.2 Preliminaries


3.3 Maximum likelihood estimation


3.4 Bayes estimation


3.5 Probabilities of large deviations of MLE and BE


3.6 Minimum L1-norm estimation








4 Sequential Inference for Some Processes Driven by Fractional Brownian


Motion


4.1 Introduction


4.2 Sequential maximum likelihood estimation


4.3 Sequential testing for simple hypothesis








5 Nonparametric Inference for Processes Driven by Fractional Brownian


Motion


5.1 Introduction


5.2 Identification for linear stochastic systems


5.3 Nonparametric estimation of trend











6 Parametric Inference for Some SDE’s Driven by Processes Related to


FBM


6.1 Introduction


6.2 Estimation of the the translation of a process driven by a fBm


6.3 Parametric inference for SDE with delay governed by a fBm


6.4 Parametric estimation for linear system of SDE driven by fBm’s with different


Hurst indices


6.5 Parametric estimation for SDE driven by mixed fBm


6.6 Alternate approach for estimation in models driven by fBm


6.7 Maximum likelihood estimation under misspecified model











7 Parametric Estimation for Processes Driven by Fractional Brownian Sheet


7.1 Introduction


7.2 Parametric estimation for linear SDE driven by a fractional Brownian sheet








8 Parametric Estimation for Processes Driven by Infinite Dimensional Fractional


Brownian Motion


8.1 Introduction


8.2 Parametric estimation for SPDE driven by infinite dimensional fBm


8.3 Parametric estimation for stochastic parabolic equations driven by infinite


dimensional fBm











9 Estimation of Self-Similarity Index


9.1 Introduction


9.2 Estimation of the Hurst index H when H is a constant and 12 < H < 1 for fBm


9.3 Estimation of scaling exponent function H(.) for locally self-similar processes











10 Filtering and Prediction for Linear Systems Driven by Fractional Brownian


Motion


10.1 Introduction


10.2 Prediction of fractional Brownian motion


10.3 Filtering in a simple linear system driven by a fBm


10.4 General approach for filtering for linear systems driven by fBm





References





Index