Statistical Inference for Fractional Diffusion Processes

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