Theory of stochastic processes : with applications to financial mathematics and risk theory / Dmytro Gusak ... [and others].

Format:

Book

Contributor:

Gusak, D. V. (Dmitriĭ Vasilʹevich)

Class of 1924 Book Fund.

Series:

Problem books in mathematics

Problem books in mathematics, 0941-3502

Language:

English

Subjects (All):

Stochastic processes.

Business mathematics.

Risk.

Physical Description:

xii, 375 pages : illustrations ; 25 cm.

Place of Publication:

New York : Springer, [2010]

Summary:

This book is a collection of exercises covering all the main topics in the modern theory of stochastic processes and its applications, including finance, actuarial mathematics, queuing theory, and risk theory.

The aim of this book is to provide the reader with the theoretical and practical material necessary for deeper understanding of the main topics in the theory of stochastic processes and its related fields.

The book is divided into chapters according to the various topics. Each chapter contains problems, hints, solutions, as well as a self-contained theoretical part which gives all the necessary material for solving the problems. References to the literature are also given.

The exercises have various levels of complexity and vary from simple ones, useful for students studying basic notions and technique, to very advanced ones that reveal some important theoretical facts and constructions.

This book is one of the largest collections of problems in the theory of stochastic processes and its applications. The problems in this book can be useful for undergraduate and graduate students, as well as for specialists in the theory of stochastic processes.

Contents:

1 Definition of stochastic process. Cylinder σ-algebra, finite-dimensional distributions, the Kolmogorov theorem 1

Theoretical grounds 1

Bibliography 3

Problems 3

Hints 7

Answers and Solutions 9

2 Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions 11

Theoretical grounds 11

Bibliography 13

Problems 13

Hints 16

Answers and Solutions 17

3 Trajectories. Modifications. Filiations 23

Theoretical grounds 21

Bibliography 24

Problems 24

Hints 29

Answers and Solutions 31

4 Continuity. Differentiability. Integrability 33

Theoretical grounds 33

Bibliography 34

Problems 34

Hints 38

Answers and Solutions 40

5 Stochastic processes with independent increments. Wiener and Poisson processes. Poisson point measures 43

Theoretical grounds 43

Bibliography 46

Problems 47

Hints 53

Answers and Solutions 54

6 Gaussian processes 59

Theoretical grounds 59

Bibliography 61

Problems 62

Hints 66

Answers and Solutions 67

7 Martingales and related processes in discrete and continuous time. stopping times 71

Theoretical grounds 71

Bibliography 79

Problems 79

Hints 93

Answers and Solutions 98

8 Stationary discrete- and continuous-time processes. Stochastic integral over measure with orthogonal values 107

Theoretical grounds 107

Bibliography 110

Problems 111

Hints 119

Answers and Solutions 122

9 Prediction and interpolation 129

Theoretical grounds 129

Bibliography 130

Problems 131

Hints 133

Answers and Solutions 135

10 Markov chains: Discrete and continuous time 137

Theoretical grounds 137

Bibliography 140

Problems 140

Hints 152

Answers and Solutions 154

11 Renewal theory. Queueing theory 159

Theoretical grounds 159

Bibliography 162

Problems 162

Hints 169

Answers and Solutions 170

12 Markov and diffusion processes 175

Theoretical grounds 175

Bibliography 182

Problems 182

Hints 186

Answers and Solutions 188

13 It ̥stochastic integral. lt ̥formula. Tanaka formula 193

Theoretical grounds 193

Bibliography 196

Problems 196

Hints 205

Answers and Solutions 209

14 Stochastic differential equations 215

Theoretical grounds 215

Bibliography 217

Problems 217

Hints 223

Answers and Solutions 225

15 Optimal stopping of random sequences and processes 229

Theoretical grounds 229

Bibliography 231

Problems 231

Hints 235

Answers and Solutions 237

16 Measures in a functional spaces. Weak convergence, probability metrics. Functional limit theorems 241

Theoretical grounds 241

Bibliography 250

Problems 250

Hints 259

Answers and Solutions 262

17 Statistics of stochastic processes 271

Theoretical grounds 271

Bibliography 281

Problems 281

Hints 286

Answers and Solutions 287

18 Stochastic processes in financial mathematics (discrete time) 303

Theoretical grounds 303

Bibliography 306

Problems 306

Hints 310

Answers and Solutions 311

19 Stochastic processes in financial mathematics (continuous time) 315

Theoretical grounds 315

Bibliography 317

Problems 317

Hints 322

Answers and Solutions 322

20 Basic functionals of the risk theory 327

Theoretical grounds 327

Bibliography 343

Problems 343

Hints 348

Answers and Solutionst 350.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1924 Book Fund.

ISBN:

9780387878614

0387878610

OCLC:

297148434