Discretization of Processes

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.    

• 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

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)

• Pierre et Marie Curie, Institut de Mathématiques, Université Paris VI -, Paris Cedex, France

  Jean Jacod

• Department of Statistics, Columbia University, New York, USA

  Philip Protter

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