DescriptionA 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.
The Stochastic Integral.
Processes with Finite Variation.
Processes with Finite Semivariation.
The Itô Formula.
Stochastic Integration in the Plane.
Two-Parameter Processes with Finite Variation.
Two-Parameter Processes with Finite Semivariation.
"...it can be expected that...just like the author's 1967 volume, this book will stimulate further research on vector stochastic integration and can serve as a graduate-level reference work." (Mathematical Reviews Issue 2001h)
"Dense, detailed, comprehensive introduction. Contains...material only found before in journals..." (American Mathematical Monthly, March 2002)
“…a highly technical book.” (The Mathematical Gazette, March 2002)
"The author of this important and interesting book is a well-known specialist on vector measures." (Zentralblatt Math, Vol.974, No. 24 2001)