Theory and Statistical Applications of Stochastic Processes

This book is concerned with the theory of stochastic processes and the theoretical aspects of statistics for stochastic processes. It combines classic topics such as construction of stochastic processes, associated filtrations, processes with independent increments, Gaussian processes, martingales, Markov properties, continuity and related properties of trajectories with contemporary subjects: integration with respect to Gaussian processes, Itô integration, stochastic analysis, stochastic differential equations, fractional Brownian motion and parameter estimation in diffusion models.

Auteur
Yuliya Mishura, National University of Kyiv, Ukraine Georgiy Shevchenko, National University of Kyiv, Ukraine

Contenu

Preface xi

Introduction xiii

Part 1 Theory of Stochastic Processes 1

Chapter 1 Stochastic Processes General Properties. Trajectories, Finite-dimensional Distributions 3

1.1 Definition of a stochastic process 3

1.2 Trajectories of a stochastic process Some examples of stochastic processes 5

1.2.1 Definition of trajectory and some examples 5

1.2.2 Trajectory of a stochastic process as a random element.8

1.3 Finite-dimensional distributions of stochastic processes: consistency conditions.10

1.3.1 Definition and properties of finite-dimensional distributions 10

1.3.2 Consistency conditions.11

1.3.3 Cylinder sets and generated -algebra 13

1.3.4 Kolmogorov theorem on the construction of a stochastic process by the family of probability distributions 15

1.4 Properties of -algebra generated by cylinder sets. The notion of -algebra generated by a stochastic process 19

Chapter 2 Stochastic Processes with Independent Increments 21

2.1 Existence of processes with independent increments in terms of incremental characteristic functions 21

2.2 Wiener process 24

2.2.1 One-dimensional Wiener process 24

2.2.2 Independent stochastic processes Multidimensional Wiener process 24

2.3 Poisson process 27

2.3.1 Poisson process defined via the existence theorem 27

2.3.2 Poisson process defined via the distributions of the increments 28

2.3.3 Poisson process as a renewal process 30

2.4 Compound Poisson process 33

2.5 Lévy processes 34

2.5.1 Wiener process with a drift 36

2.5.2 Compound Poisson process as a Lévy process 36

2.5.3 Sum of a Wiener process with a drift and a Poisson process 36

2.5.4 Gamma process 37

2.5.5 Stable Lévy motion37

2.5.6 Stable Lévy subordinator with stability parameter (0, 1) 38

Chapter 3 Gaussian Processes Integration with Respect to Gaussian Processes 39

3.1 Gaussian vectors 39

3.2 Theorem of Gaussian representation (theorem on normal correlation) 42

3.3 Gaussian processes. 44

3.4 Examples of Gaussian processes 46

3.4.1 Wiener process as an example of a Gaussian process 46

3.4.2 Fractional Brownian motion.48

3.4.3 Sub-fractional and bi-fractional Brownian motion 50

3.4.4 Brownian bridge 50

3.4.5 OrnsteinUhlenbeck process 51

3.5 Integration of non-random functions with respect to Gaussian processes 52

3.5.1 General approach 52

3.5.2 Integration of non-random functions with respect to the Wiener process 54

3.5.3 Integration w.r.t the fractional Brownian motion 57

3.6 Two-sided Wiener process and fractional Brownian motion: Mandelbrotvan Ness representation of fractional Brownian motion 60

3.7 Representation of fractional Brownian motion as the Wiener integral on the compact integral 63

Chapter 4 Construction, Properties and Some Functionals of the Wiener Process and Fractional Brownian Motion 67

4.1 Construction of a Wiener process on the interval [0, 1] 67

4.2 Construction of a Wiener process on R+ 72

4.3 Nowhere differentiability of the trajectories of a Wiener process 74

4.4 Power variation of the Wiener process and of the fractional Brownian motion77

4.4.1 Ergodic theorem for power variations 77

4.5 Self-similar stochastic processes 79

4.5.1 Definition of self-similarity and some examples 79

4.5.2 Power variations of self-similar processes on finite intervals.80

Chapter 5 Martingales and Related Processes 85

5.1 Notion of stochastic basis with filtration 85

5.2 Notion of (sub-, super-) martingale: elementary properties 86

5.3 Examples of (sub-, super-) martingales 87

5.4 Markov moments and stopping times 90

5.5 Martingales and related processes with discrete time 96

5.5.1 Upcrossings of the interval and existence of the limit of submartingale 96 5.5.2 Examples of martingales having...