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.
Зміст
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 f Bm
1.5 Fractional Ornstein-Uhlenbeck type process
1.6 Mixed fractional Brownian motion
1.7 Donsker type approximation for f Bm with Hurst index H >
12
1.8 Simulation of fractional Brownian motion
1.9 Remarks on application of modelling by f Bm in mathematical
finance
1.10 Path wise integration with respect to f Bm
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
f Bm
6.3 Parametric inference for SDE with delay governed by a
f Bm
6.4 Parametric estimation for linear system of SDE driven by
f Bm’s with different
Hurst indices
6.5 Parametric estimation for SDE driven by mixed f Bm
6.6 Alternate approach for estimation in models driven by
f Bm
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 f Bm
8.3 Parametric estimation for stochastic parabolic equations
driven by infinite
dimensional f Bm
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 f Bm
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 f Bm
10.4 General approach for filtering for linear systems driven by
f Bm
References
Index
Про автора
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.
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