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3023 results in Probability theory and stochastic processes

Page 86 of 152

  • Preface

  • Daniel W. Stroock, Massachusetts Institute of Technology
  • Book:
    Probability Theory
    Published online:
    05 June 2012
    Print publication:
    31 December 2010, pp xv-xxii

  • Summary

    From the Preface to the First Edition

    When writing a graduate level mathematics book during the last decade of the twentieth century, one probably ought not inquire too closely into one's motivation. In fact, if ones own pleasure from the exercise is not suffcient to justify the effort, then one should seriously consider dropping the project. Thus, to those who (either before or shortly after opening it) ask for whom was this book written, my pale answer is me; and, for this reason, I thought that I should preface this preface with an explanation of who I am and what were the peculiar educational circumstances that eventually gave rise to this somewhat peculiar book.

    My own introduction to probability theory began with a private lecture from H.P. McKean, Jr. At the time, I was a (more accurately, the) graduate student of mathematics at what was then called The Rockefeller Institute for Biological Sciences. My offcial mentor there was M. Kac, whom I had cajoled into becoming my adviser after a year during which I had failed to insert even one micro-electrode into the optic nerves of innumerable limuli. However, as I soon came to realize, Kac had accepted his role on the condition that it would not become a burden. In particular, he had no intention of wasting much of his own time on a reject from the neurophysiology department.

  • Chapter 8 - Gaussian Measures on a Banach Space

  • Daniel W. Stroock, Massachusetts Institute of Technology
  • Book:
    Probability Theory
    Published online:
    05 June 2012
    Print publication:
    31 December 2010, pp 299-366

  • Summary

    As I said at the end of § 4.3.2, the distribution of Brownian motion is called Wiener measure because Wiener was the first to construct it. Wiener's own thinking about his measure had little or nothing in common with the Lévy–Khinchine program. Instead, he looked upon his measure as a Gaussian measure on an infinite dimensional space, and most of what he did with his measure is best understood from that perspective. Thus, in this chapter, we will look at Wiener measure from a strictly Gaussian point of view. More generally, we will be dealing here with measures on a real Banach space E that are centered Gaussian in the sense that, for each x* in the dual space E*, x ∈ E ↦ 〈x, x*〉, ∈ ℝ is a centered Gaussian random variable. Not surprisingly, such a measure will be said to be a centered Gaussian measure on E.

    Although the ideas that I will use are already implicit in Wiener's work, it was I. Segal and his school, especially L. Gross, who gave them the form presented here.

    The Classical Wiener Space

    In order to motivate what follows, it is helpful to first understand Wiener measure from the point of view which I will be adopting here.

Probability and Statistics by Example
  • Volume 2, Markov Chains: A Primer in Random Processes and their Applications
  • Yuri Suhov, Mark Kelbert
  • Published online:
    11 November 2010
    Print publication:
    24 April 2008
  • Probability and Statistics are as much about intuition and problem solving as they are about theorem proving. Because of this, students can find it very difficult to make a successful transition from lectures to examinations to practice, since the problems involved can vary so much in nature. Since the subject is critical in many modern applications such as mathematical finance, quantitative management, telecommunications, signal processing, bioinformatics, as well as traditional ones such as insurance, social science andengineering, the authors have rectified deficiencies in traditional lecture-based methods by collecting together a wealth of exercises with complete solutions, adapted to needs and skills of students. Following on from the success of Probability and Statistics by Example: Basic Probability and Statistics, the authors here concentrate on random processes, particularly Markov processes, emphasising modelsrather than general constructions. Basic mathematical facts are supplied as and when they are needed andhistorical information is sprinkled throughout.

Probability and Statistics by Example
  • Volume 1, Basic Probability and Statistics
  • Yuri Suhov, Mark Kelbert
  • Published online:
    11 November 2010
    Print publication:
    13 October 2005
  • Probability and Statistics are as much about intuition and problem solving, as they are about theorem proving. Because of this, students can find it very difficult to make a successful transition from lectures to examinations to practice, since the problems involved can vary so much in nature. Since the subject is critical in many modern applications such as mathematical finance, quantitative management, telecommunications, signal processing, bioinformatics, as well as traditional ones such as insurance, social science and engineering, the authors have rectified deficiencies in traditional lecture-based methods by collecting together a wealth of exercises for which they have supplied complete solutions. These solutions are adapted to needs and skills of students. To make it of broad value, the authors supply basic mathematical facts as and when they are needed, and have sprinkled some historical information throughout the text.

Ranks of Elliptic Curves and Random Matrix Theory
  • Edited by J. B. Conrey, D. W. Farmer, F. Mezzadri, N. C. Snaith
  • Published online:
    10 November 2010
    Print publication:
    08 February 2007
  • Random matrix theory is an area of mathematics first developed by physicists interested in the energy levels of atomic nuclei, but it can also be used to describe some exotic phenomena in the number theory of elliptic curves. The purpose of this book is to illustrate this interplay of number theory and random matrices. It begins with an introduction to elliptic curves and the fundamentals of modelling by a family of random matrices, and moves on to highlight the latest research. There are expositions of current research on ranks of elliptic curves, statistical properties of families of elliptic curves and their associated L-functions and the emerging uses of random matrix theory in this field. Most of the material here had its origin in a Clay Mathematics Institute workshop on this topic at the Newton Institute in Cambridge and together these contributions provide a unique in-depth treatment of the subject.

Stochastic Partial Differential Equations with Lévy Noise
  • An Evolution Equation Approach
  • S. Peszat, J. Zabczyk
  • Published online:
    10 November 2010
    Print publication:
    11 October 2007
  • Recent years have seen an explosion of interest in stochastic partial differential equations where the driving noise is discontinuous. In this comprehensive monograph, two leading experts detail the evolution equation approach to their solution. Most of the results appeared here for the first time in book form. The authors start with a detailed analysis of Lévy processes in infinite dimensions and their reproducing kernel Hilbert spaces; cylindrical Lévy processes are constructed in terms of Poisson random measures; stochastic integrals are introduced. Stochastic parabolic and hyperbolic equations on domains of arbitrary dimensions are studied, and applications to statistical and fluid mechanics and to finance are also investigated. Ideal for researchers and graduate students in stochastic processes and partial differential equations, this self-contained text will also interest those working on stochastic modeling in finance, statistical physics and environmental science.

Page 86 of 152