Theory and Statistical Applications of Stochastic Processes [Hardcover]

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, Itȏ integration, stochastic analysis, stochastic differential equations, fractional Brownian motion and parameter estimation in diffusion models.

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

Chapter 2 Stochastic Processes with Independent Increments 21

2.1 Existence of processes with independent increments in termsl³’