Theory of Stochastic Processes

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About this book

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.

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Overview

Stochastic Processes in the Decades after 1950

Mathematical background on stochastic processes

Keywords

• Gaussian process
• Markov chain
• Martingale
• Poisson process
• Stochastic Differential Equations
• Stochastic Processes
• diffusion process
• filtration
• finite-dimensional distribution
• queueing theory
• renewal theory
• stochastic process

Table of contents (20 chapters)

Front Matter

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

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

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

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Trajectories. Modifications. Filtrations

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Continuity. Differentiability. Integrability

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

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

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Gaussian processes

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Martingales and related processes in discrete and continuous time. Stopping times

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

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

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Prediction and interpolation

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Markov chains: Discrete and continuous time

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Renewal theory. Queueing theory

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Markov and diffusion processes

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Itô stochastic integral. Itô formula. Tanaka formula

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Stochastic differential equations

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Optimal stopping of random sequences and processes

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

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

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Statistics of stochastic processes

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Stochastic processes in financial mathematics (discrete time)

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Stochastic processes in financial mathematics (continuous time)

• Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

Reviews

From the reviews:

“Chapter deals with the statistics of stochastic processes, mainly hypotheses testing, a relatively uncommon subject. … The major strength of this problem book is the breadth and depth of coverage that five experts in their respective subfields condensed in only 375 pages. … the book is a valuable addition to the literature on stochastic processes. … any course in stochastics at the advanced undergraduate or beginning to intermediate graduate level is almost sure to interest its table of contents substantially.” (Giuseppe Castellacci, Mathematical Reviews, Issue 2011 f)

“Advanced undergraduates and postgraduates in mathematics, and teaching staff at these levels. This is a book in the Springer series on Problem Books in Mathematics, presenting a series of problems … . Each of the 20 chapters in this book has a condensed outline of the topic being considered, a bibliography, the problems, and then hints or solutions to most of the problems.” (David J. Hand, International Statistical Review, Vol. 78 (3), 2010)

“This book provides a collection of more than 800 problems for the theory of stochastic processes. It is divided into 20 chapters that cover different aspects of this theory. … this compilation is new in its broadness and completeness for the theory of stochastic processes and is well suited for students in their self-studies as well as lecturers to prepare their classes in this field of probability theory.” (Claudia Hein, Zentralblatt MATH, Vol. 1189, 2010)

“Each chapter consists of a brief review of theory followed by … a list of problems, hints (for the solution of) pertaining to most of the problems in the chapter, and a section giving ‘Answers and Solutions’ for many but not necessarily all problems. … It might also be used in seminars or in advanced topics courses. … There is also a set of graphical representations of various stochastic processes. … an excellent contribution and anyone who works through the problems will be well rewarded.” (Donald E. Myers, Technometrics, Vol. 53 (3), August, 2011)

Authors and Affiliations

Inst. Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

University of Kiev, Dept. Mechanics and Mathematics, National Taras Shevchenko, Kiev, Ukraine

Bibliographic Information

• Book Title : Theory of Stochastic Processes
• Book Subtitle : With Applications to Financial Mathematics and Risk Theory
• Authors : Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko
• Series Title : Problem Books in Mathematics
• DOI : https://doi.org/10.1007/978-0-387-87862-1
• Publisher : Springer New York, NY
• eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)
• Copyright Information : Springer-Verlag New York 2010
• Hardcover ISBN : 978-0-387-87861-4 Published: 04 December 2009
• Softcover ISBN : 978-1-4614-2506-9 Published: 03 May 2012
• eBook ISBN : 978-0-387-87862-1 Published: 10 July 2010
• Series ISSN : 0941-3502
• Series E-ISSN : 2197-8506
• Edition Number : 1
• Number of Pages : XII, 376
• Number of Illustrations : 8 b/w illustrations
• Topics : Probability Theory and Stochastic Processes , Actuarial Sciences , Business Mathematics , Statistics for Business, Management, Economics, Finance, Insurance