An Introduction To Infinite Dimensional Analysis

Based on well-known lectures given at Scuola Normale Superiore in Pisa, this book introduces analysis in a separable Hilbert space of infinite dimension. It starts from the definition of Gaussian measures in Hilbert spaces, concepts such as the Cameron-Martin formula, Brownian motion and Wiener integral are introduced in a simple way. These concepts are then used to illustrate basic stochastic dynamical systems and Markov semi-groups, paying attention to their long-time behavior.

This text was born out of an advanced mathematical economics seminar at Caltech in 1989-90. We realized that the typical graduate student in mathematical economics has to be familiar with a vast amount of material that spans several traditional fields in mathematics. Much of the mate rial appears only in esoteric research monographs that are designed for specialists, not for the sort of generalist that our students need be. We hope that in a small way this text will make the material here accessible to a much broader audience. While our motivation is to present and orga nize the analytical foundations underlying modern economics and finance, this is a book of mathematics, not of economics. We mention applications to economics but present very few of them. They are there to convince economists that the material has so me relevance and to let mathematicians know that there are areas of application for these results. We feel that this text could be used for a course in analysis that would benefit math ematicians, engineers, and scientists. Most of the material we present is available elsewhere, but is scattered throughout a variety of sources and occasionally buried in obscurity. Some of our results are original (or more likely, independent rediscoveries). We have included some material that we cannot honestly say is neces sary to understand modern economic theory, but may yet prove useful in future research.

The infinite dimensional analysis as a branch of mathematical sciences was formed in the late 19th and early 20th centuries. Motivated by problems in mathematical physics, the first steps in this field were taken by V. Volterra, R. GateallX, P. Levy and M. Frechet, among others (see the preface to Levy[2]). Nevertheless, the most fruitful direction in this field is the infinite dimensional integration theory initiated by N. Wiener and A. N. Kolmogorov which is closely related to the developments of the theory of stochastic processes. It was Wiener who constructed for the first time in 1923 a probability measure on the space of all continuous functions (i. e. the Wiener measure) which provided an ideal math ematical model for Brownian motion. Then some important properties of Wiener integrals, especially the quasi-invariance of Gaussian measures, were discovered by R. Cameron and W. Martin[l, 2, 3]. In 1931, Kolmogorov[l] deduced a second partial differential equation for transition probabilities of Markov processes order with continuous trajectories (i. e. diffusion processes) and thus revealed the deep connection between theories of differential equations and stochastic processes. The stochastic analysis created by K. Ito (also independently by Gihman [1]) in the forties is essentially an infinitesimal analysis for trajectories of stochastic processes. By virtue of Ito's stochastic differential equations one can construct diffusion processes via direct probabilistic methods and treat them as function als of Brownian paths (i. e. the Wiener functionals).

This book presents functional analytic methods in a unified manner with applications to economics, social sciences, and engineering. Ideal for those without an extensive background in the area, it develops topology, convexity, Banach lattices, integration, correspondences, and the analytic approach to Markov processes. Many of the results were previously available only in esoteric monographs and will interest researchers and students who will find the material readily applicable to problems in control theory and economics.

Over the past six decades, several extremely important fields in mathematics have been developed. Among these are Itô calculus, Gaussian measures on Banach spaces, Malliavan calculus, and white noise distribution theory. These subjects have many applications, ranging from finance and economics to physics and biology. Unfortunately, the background information required to conduct research in these subjects presents a tremendous roadblock. The background material primarily stems from an abstract subject known as infinite dimensional topological vector spaces. While this information forms the backdrop for these subjects, the books and papers written about topological vector spaces were never truly written for researchers studying infinite dimensional analysis. Thus, the literature for topological vector spaces is dense and difficult to digest, much of it being written prior to the 1960s. Tools for Infinite Dimensional Analysis aims to address these problems by providing an introduction to the background material for infinite dimensional analysis that is friendly in style and accessible to graduate students and researchers studying the above-mentioned subjects. It will save current and future researchers countless hours and promote research in these areas by removing an obstacle in the path to beginning study in areas of infinite dimensional analysis. Features Focused approach to the subject matter Suitable for graduate students as well as researchers Detailed proofs of primary results

This book presents the mathematical issues that arise in modeling the interest rate term structure by casting the interest-rate models as stochastic evolution equations in infinite dimensions. The text includes a crash course on interest rates, a self-contained introduction to infinite dimensional stochastic analysis, and recent results in interest rate theory. From the reviews: "A wonderful book. The authors present some cutting-edge math." --WWW.RISKBOOK.COM

Over the past six decades, several extremely important fields in mathematics have been developed. Among these are Itô calculus, Gaussian measures on Banach spaces, Malliavan calculus, and white noise distribution theory. These subjects have many applications, ranging from finance and economics to physics and biology. Unfortunately, the background information required to conduct research in these subjects presents a tremendous roadblock. The background material primarily stems from an abstract subject known as infinite dimensional topological vector spaces. While this information forms the backdrop for these subjects, the books and papers written about topological vector spaces were never truly written for researchers studying infinite dimensional analysis. Thus, the literature for topological vector spaces is dense and difficult to digest, much of it being written prior to the 1960s. Tools for Infinite Dimensional Analysis aims to address these problems by providing an introduction to the background material for infinite dimensional analysis that is friendly in style and accessible to graduate students and researchers studying the above-mentioned subjects. It will save current and future researchers countless hours and promote research in these areas by removing an obstacle in the path to beginning study in areas of infinite dimensional analysis. Features Focused approach to the subject matter Suitable for graduate students as well as researchers Detailed proofs of primary results