A breakthrough approach to the theory and applications of stochastic integration The theory of stochastic integration has become an intensely studied topic in recent years, owing to its extraordinarily successful application to financial mathematics, stochastic differential equations, and more. This book features a new measure theoretic approach to stochastic integration, opening up the field for researchers in measure and integration theory, functional analysis, probability theory, and stochastic processes. World-famous expert on vector and stochastic integration in Banach spaces Nicolae Dinculeanu compiles and consolidates information from disparate journal articles-including his own results-presenting a comprehensive, up-to-date treatment of the theory in two major parts. He first develops a general integration theory, discussing vector integration with respect to measures with finite semivariation, then applies the theory to stochastic integration in Banach spaces. Vector Integration and Stochastic Integration in Banach Spaces goes far beyond the typical treatment of the scalar case given in other books on the subject. Along with such applications of the vector integration as the Reisz representation theorem and the Stieltjes integral for functions of one or two variables with finite semivariation, it explores the emergence of new classes of summable processes that make applications possible, including square integrable martingales in Hilbert spaces and processes with integrable variation or integrable semivariation in Banach spaces. Numerous references to existing results supplement this exciting, breakthrough work.

This is a development of the book entitled Multidimensional Second Order Stochastic Processes. It provides a research expository treatment of infinite-dimensional stationary and nonstationary stochastic processes or time series, based on Hilbert and Banach space-valued second order random variables. Stochastic measures and scalar or operator bimeasures are fully discussed to develop integral representations of various classes of nonstationary processes such as harmonizable, V-bounded, Cramér and Karhunen classes as well as the stationary class. A new type of the Radon-Nikodým derivative of a Banach space-valued measure is introduced, together with Schauder basic measures, to study uniformly bounded linearly stationary processes.Emphasis is on the use of functional analysis and harmonic analysis as well as probability theory. Applications are made from the probabilistic and statistical points of view to prediction problems, Kalman filter, sampling theorems and strong laws of large numbers. Generalizations are made to consider Banach space-valued stochastic processes to include processes of pth order for p ≥ 1. Readers may find that the covariance kernel is always emphasized and reveals another aspect of stochastic processes.This book is intended not only for probabilists and statisticians, but also for functional analysts and communication engineers.

The main goal of this Handbook isto survey measure theory with its many different branches and itsrelations with other areas of mathematics. Mostly aggregating many classical branches of measure theory the aim of the Handbook is also to cover new fields, approaches and applications whichsupport the idea of "measure" in a wider sense, e.g. the ninth part of the Handbook. Although chapters are written of surveys in the variousareas they contain many special topics and challengingproblems valuable for experts and rich sources of inspiration.Mathematicians from other areas as well as physicists, computerscientists, engineers and econometrists will find useful results andpowerful methods for their research. The reader may find in theHandbook many close relations to other mathematical areas: realanalysis, probability theory, statistics, ergodic theory,functional analysis, potential theory, topology, set theory,geometry, differential equations, optimization, variationalanalysis, decision making and others. The Handbook is a richsource of relevant references to articles, books and lecturenotes and it contains for the reader's convenience an extensivesubject and author index.

A breakthrough approach to the theory and applications of stochastic integration The theory of stochastic integration has become an intensely studied topic in recent years, owing to its extraordinarily successful application to financial mathematics, stochastic differential equations, and more. This book features a new measure theoretic approach to stochastic integration, opening up the field for researchers in measure and integration theory, functional analysis, probability theory, and stochastic processes. World-famous expert on vector and stochastic integration in Banach spaces Nicolae Dinculeanu compiles and consolidates information from disparate journal articles-including his own results-presenting a comprehensive, up-to-date treatment of the theory in two major parts. He first develops a general integration theory, discussing vector integration with respect to measures with finite semivariation, then applies the theory to stochastic integration in Banach spaces. Vector Integration and Stochastic Integration in Banach Spaces goes far beyond the typical treatment of the scalar case given in other books on the subject. Along with such applications of the vector integration as the Reisz representation theorem and the Stieltjes integral for functions of one or two variables with finite semivariation, it explores the emergence of new classes of summable processes that make applications possible, including square integrable martingales in Hilbert spaces and processes with integrable variation or integrable semivariation in Banach spaces. Numerous references to existing results supplement this exciting, breakthrough work.

An increasing number of statistical problems and methods involve infinite-dimensional aspects. This is due to the progress of technologies which allow us to store more and more information while modern instruments are able to collect data much more effectively due to their increasingly sophisticated design. This evolution directly concerns statisticians, who have to propose new methodologies while taking into account such high-dimensional data (e.g. continuous processes, functional data, etc.). The numerous applications (micro-arrays, paleo- ecological data, radar waveforms, spectrometric curves, speech recognition, continuous time series, 3-D images, etc.) in various fields (biology, econometrics, environmetrics, the food industry, medical sciences, paper industry, etc.) make researching this statistical topic very worthwhile. This book gathers important contributions on the functional and operatorial statistics fields.

Considering Poisson random measures as the driving sources for stochastic (partial) differential equations allows us to incorporate jumps and to model sudden, unexpected phenomena. By using such equations the present book introduces a new method for modeling the states of complex systems perturbed by random sources over time, such as interest rates in financial markets or temperature distributions in a specific region. It studies properties of the solutions of the stochastic equations, observing the long-term behavior and the sensitivity of the solutions to changes in the initial data. The authors consider an integration theory of measurable and adapted processes in appropriate Banach spaces as well as the non-Gaussian case, whereas most of the literature only focuses on predictable settings in Hilbert spaces. The book is intended for graduate students and researchers in stochastic (partial) differential equations, mathematical finance and non-linear filtering and assumes a knowledge of the required integration theory, existence and uniqueness results and stability theory. The results will be of particular interest to natural scientists and the finance community. Readers should ideally be familiar with stochastic processes and probability theory in general, as well as functional analysis and in particular the theory of operator semigroups. ​

This volume contains a selection of articles on the theme "vector measures, integration and applications" together with some related topics. The articles consist of both survey style and original research papers, are written by experts in thearea and present a succinct account of recent and up-to-date knowledge. The topic is interdisciplinary by nature and involves areas such as measure and integration (scalar, vector and operator-valued), classical and harmonic analysis, operator theory, non-commutative integration, andfunctional analysis. The material is of interest to experts, young researchers and postgraduate students.

This extraordinary compilation is an expansion of the recent American Mathematical Society Special Session celebrating M. M. Rao's distinguished career and includes most of the presented papers as well as ancillary contributions from session invitees. This book shows the effectiveness of abstract analysis for solving fundamental problems of stochas

The 1990 Seminar on Stochastic Processes was held at the University of British Columbia from May 10 through May 12, 1990. This was the tenth in a series of annual meetings which provide researchers with the opportunity to discuss current work on stochastic processes in an informal and enjoyable atmosphere. Previous seminars were held at Northwestern University, Princeton University, the Univer sity of Florida, the University of Virginia and the University of California, San Diego. Following the successful format of previous years, there were five invited lectures, delivered by M. Marcus, M. Vor, D. Nualart, M. Freidlin and L. C. G. Rogers, with the remainder of the time being devoted to informal communications and workshops on current work and problems. The enthusiasm and interest of the participants created a lively and stimulating atmosphere for the seminar. A sample of the research discussed there is contained in this volume. The 1990 Seminar was made possible by the support of the Natural Sciences and Engin~ring Research Council of Canada, the Southwest University Mathematics Society of British Columbia, and the University of British Columbia. To these entities and the organizers of this year's conference, Ed Perkins and John Walsh, we extend oul' thanks. Finally, we acknowledge the support and assistance of the staff at Birkhauser Boston.

Providing an introduction to stochastic optimal control in infinite dimension, this book gives a complete account of the theory of second-order HJB equations in infinite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results (e.g. the dynamic programming principle) with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in infinite-dimensional stochastic control, via BSDEs. The book is of interest to both pure and applied researchers working in the control theory of stochastic PDEs, and in PDEs in infinite dimension. Readers from other fields who want to learn the basic theory will also find it useful. The prerequisites are: standard functional analysis, the theory of semigroups of operators and its use in the study of PDEs, some knowledge of the dynamic programming approach to stochastic optimal control problems in finite dimension, and the basics of stochastic analysis and stochastic equations in infinite-dimensional spaces.