Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory / Edition 1

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

Overview

This book is a collection of exercises covering all the main topics in the modern theory of shastic 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 shastic 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 shastic 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 shastic processes.

Product Details

ISBN-13:	9780387878614
Publisher:	Springer New York
Publication date:	12/04/2009
Series:	Problem Books in Mathematics
Edition description:	2010
Pages:	376
Product dimensions:	6.40(w) x 9.30(h) x 1.10(d)

Table of Contents

Definition of shastic process. Cylinder #x03C3;-algebra, finite-dimensional distributions, the Kolmogorov theorem.- Characteristics of a shastic process. Mean and covariance functions. Characteristic functions.- Trajectories. Modifications. Filtrations.- Continuity. Differentiability. Integrability.- Shastic processes with independent increments. Wiener and Poisson processes. Poisson point measures.- Gaussian processes.- Martingales and related processes in discrete and continuous time. Stopping times.- Stationary discrete- and continuous-time processes. Shastic integral over measure with orthogonal values.- Prediction and interpolation.- Markov chains: Discrete and continuous time.- Renewal theory. Queueing theory.- Markov and diffusion processes.- It#x00F4; shastic integral. It#x00F4; formula. Tanaka formula.- Shastic differential equations.- Optimal stopping of random sequences and processes.- Measures in a functional spaces. Weak convergence, probability metrics.Functional limit theorems.- Statistics of shastic processes.- Shastic processes in financial mathematics (discrete time).- Shastic processes in financial mathematics (continuous time).- Basic functionals of the risk theory.

Editorial Reviews

From the reviews:

“Chapter deals with the statistics of shastic 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 shastic processes. … any course in shastics 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 shastic 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 shastic 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 shastic processes. … an excellent contribution and anyone who works through the problems will be well rewarded.” (Donald E. Myers, Technometrics, Vol. 53 (3), August, 2011)

From the Publisher