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© 2012

Discretization of Processes

Benefits

  • The first and so far the only book in this area

  • Presents the important results in a coherent and unified manner

  • Includes systematic, creative and original ways to use sophisticated (but highly technical) tools

Book

Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 67)

Table of contents

  1. Front Matter
    Pages I-XIV
  2. Introduction and Preliminary Material

    1. Front Matter
      Pages 1-1
    2. Jean Jacod, Philip Protter
      Pages 3-21
    3. Jean Jacod, Philip Protter
      Pages 23-59
  3. The Basic Results

    1. Front Matter
      Pages 61-61
    2. Jean Jacod, Philip Protter
      Pages 63-96
    3. Jean Jacod, Philip Protter
      Pages 97-123
    4. Jean Jacod, Philip Protter
      Pages 125-185
    5. Jean Jacod, Philip Protter
      Pages 187-212
  4. More Laws of Large Numbers

    1. Front Matter
      Pages 213-213
    2. Jean Jacod, Philip Protter
      Pages 215-226
    3. Jean Jacod, Philip Protter
      Pages 227-246
    4. Jean Jacod, Philip Protter
      Pages 247-270
  5. Extensions of the Central Limit Theorems

    1. Front Matter
      Pages 271-271
    2. Jean Jacod, Philip Protter
      Pages 273-296
    3. Jean Jacod, Philip Protter
      Pages 371-426
  6. Various Extensions

    1. Front Matter
      Pages 427-427
    2. Jean Jacod, Philip Protter
      Pages 429-460

About this book

Introduction

In applications, and especially in mathematical finance, random time-dependent events are often modeled as stochastic processes. Assumptions are made about the structure of such processes, and serious researchers will want to justify those assumptions through the use of data.  As statisticians are wont to say, “In God we trust; all others must bring data.”
 
This book establishes the theory of how to go about estimating not just scalar parameters about a proposed model, but also the underlying structure of the model itself.  Classic statistical tools are used: the law of large numbers, and the central limit theorem. Researchers have recently developed creative and original methods to use these tools in sophisticated (but highly technical) ways to reveal new details about the underlying structure. For the first time in book form, the authors present these latest techniques, based on research from the last 10 years. They include new findings.

 
This book will be of special interest to researchers, combining the theory of mathematical finance with its investigation using market data, and it will also prove to be useful in a broad range of applications, such as to mathematical biology, chemical engineering, and physics.

Keywords

60F05, 60G44, 60H10, 60H35, 60J75, 60G51, 60G57 asymptotic error central limit theorem for stochastic processes density forecasting estimation jump processes law of large numbers for stochastic processes semimartingale stable convergence stochastic processes time-varying correlation volatility weak convergence white noise

Authors and affiliations

  1. 1.Pierre et Marie Curie, Institut de MathématiquesUniversité Paris VI -Paris CedexFrance
  2. 2.Department of StatisticsColumbia UniversityNew YorkUSA

Bibliographic information

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Reviews

From the reviews:

“It is clearly statistically oriented and intended to help practitioners to answer questions about an observed random process X. … The book may be considered as the outcome of several decades of intensive work on the statistics of semimartingales, and a large part of the stated results is due to the authors. For both theoreticians and practitioners in the vast realm of random processes, this will be an indispensable reference book.” (Dominique Lépingle, Mathematical Reviews, January, 2013)

“This new book develops a theory of limit theorems for discretely observed Itô semimartingales with a view towards statistical applications. … This monograph by two leading experts in the field of stochastic processes will certainly become a standard reference when statistical questions in semimartingale models need to be investigated. The text is very well written and is without doubt a must have for scientists interested in applications of advanced stochastic process models.” (H. M. Mai, Zentralblatt MATH, Vol. 1259, 2013)