Stochastic differential equations whose solutions are diffusion (or other random) processes have been the subject of lively mathematical research since the pioneering work of Gihman, Ito and others in the early fifties. As it gradually became clear that a great number of real phenomena in control theory, physics, biology, economics and other areas could be modelled by differential equations with stochastic perturbation terms, this research became somewhat feverish, with the results that a) the number of theroretical papers alone now numbers several hundred and b) workers interested in the field (especially from an applied viewpoint) have had no opportunity to consult a systematic account. This monograph, written by two of the world's authorities on prob ability theory and stochastic processes, fills this hiatus by offering the first extensive account of the calculus of random differential equations de fined in terms of the Wiener process. In addition to systematically ab stracting most of the salient results obtained thus far in the theory, it includes much new material on asymptotic and stability properties along with a potentially important generalization to equations defined with the aid of the so-called random Poisson measure whose solutions possess jump discontinuities. Although this monograph treats one of the most modern branches of applied mathematics, it can be read with profit by anyone with a knowledge of elementary differential equations armed with a solid course in stochastic processes from the measure-theoretic point of view."

Integration in function spaces arose in probability theory when a gen eral theory of random processes was constructed. Here credit is cer tainly due to N. Wiener, who constructed a measure in function space, integrals-with respect to which express the mean value of functionals of Brownian motion trajectories. Brownian trajectories had previously been considered as merely physical (rather than mathematical) phe nomena. A. N. Kolmogorov generalized Wiener's construction to allow one to establish the existence of a measure corresponding to an arbitrary random process. These investigations were the beginning of the development of the theory of stochastic processes. A considerable part of this theory involves the solution of problems in the theory of measures on function spaces in the specific language of stochastic pro cesses. For example, finding the properties of sample functions is connected with the problem of the existence of a measure on some space; certain problems in statistics reduce to the calculation of the density of one measure w. r. t. another one, and the study of transformations of random processes leads to the study of transformations of function spaces with measure. One must note that the language of probability theory tends to obscure the results obtained in these areas for mathematicians working in other fields. Another dir, ection leading to the study of integrals in function space is the theory and application of differential equations. A. N."

The object of queueing theory (or the theory of mass service) is the investigation of stochastic processes of a special form which are called queueing (or service) processes in this book. Two approaches to the definition of these processes are possible depending on the direction of investigation. In accordance with this fact, the exposition of the subject can be broken up into two self-contained parts. The first of these forms the content of this monograph. . The definition of the queueing processes (systems) to be used here is dose to the traditional one and is connected with the introduction of so-called governing random sequences. We will introduce algorithms which describe the governing of a system with the aid of such sequences. Such a definition inevitably becomes rather qualitative since under these conditions a completely formal construction of a stochastic process uniquely describing the evolution of the system would require introduction of a complicated phase space not to mention the difficulties of giving the distribution of such a process on this phase space.

to Mathematical Statistics Translated from the German by Kenneth Wickwire Springer-Verlag Berlin Heidelberg New York 1974 Leopold Schmetterer Professor of Statistics and Mathematics at the University of Vienna Translator: Kenneth Wickwire Department of Mathematics, University of Manchester Title of the German Original Edition: Einfiihrung in die mathematische Statistik, 2. verbesserte und wesentlich erweiterte Auflage Springer-Verlag Wien New York 1966 With 11 figures AMS Subject Classification (1970): 62-01, 62 Axx, 62 Bxx, 62 Cxx, 62D03, 62 Exx, 62 Fxx, 62 Gxx, 62 Hxx ISBN-13: 978-3-642-65544-9 e-ISBN-13: 978-3-642-65542-5 DOl: 10. 1007/978-3-642-65542-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. (c) by Springer-Verlag Berlin. Heidelberg 1974. Library of Congress Catalog Card Number 73-15290. Softcover reprint of the hardcover 1 st edition 1974 Bookbinding: Konrad Triltsch, Wiirzburg. Preface I have used the opportunity of the second edition of the German version being translated into English to alter and improve some details. Of course I tried to correct misprints and errata of the original version. Moreover some proofs have been slightly changed and I hope thereby improved."