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Theory of Stochastic Processes

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Table of Contents

Definition of stochastic process. Cylinder #x03C3;-algebra, finite-dimensional distributions, the Kolmogorov theorem.- Characteristics of a stochastic process. Mean and covariance functions. Characteristic functions.- Trajectories. Modifications. Filtrations.- Continuity. Differentiability. Integrability.- Stochastic 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. Stochastic 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; stochastic integral. It#x00F4; formula. Tanaka formula.- Stochastic differential equations.- Optimal stopping of random sequences and processes.- Measures in a functional spaces. Weak convergence, probability metrics. Functional limit theorems.- Statistics of stochastic processes.- Stochastic processes in financial mathematics (discrete time).- Stochastic processes in financial mathematics (continuous time).- Basic functionals of the risk theory.


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)

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