Change of Time and Change of Measure

Change of Time and Change of Measure provides a comprehensive account of two topics that are of particular significance in both theoretical and applied stochastics: random change of time and change of probability law.

Random change of time is key to understanding the nature of various stochastic processes, and gives rise to interesting mathematical results and insights of importance for the modeling and interpretation of empirically observed dynamic processes. Change of probability law is a technique for solving central questions in mathematical finance, and also has a considerable role in insurance mathematics, large deviation theory, and other fields.

The book comprehensively collects and integrates results from a number of scattered sources in the literature and discusses the importance of the results relative to the existing literature, particularly with regard to mathematical finance.

In this Second Edition a Chapter 13 entitled 'A Wider View' has been added. This outlines some of the developments that have taken place in the area of Change of Time and Change of Measure since the publication of the First Edition. Most of these developments have their root in the study of the Statistical Theory of Turbulence rather than in Financial Mathematics and Econometrics, and they form part of the new research area termed 'Ambit Stochastics'.

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Contents:Random Change of TimeIntegral Representations and Change of Time in Stochastic IntegralsSemimartingales: Basic Notions, Structures, Elements of Stochastic AnalysisStochastic Exponential and Stochastic Logarithm. Cumulant ProcessesProcesses with Independent Increments. Lévy ProcessesChange of Measure. General FactsChange of Measure in Models Based on Lévy ProcessesChange of Time in Semimartingale Models and Models Based on Brownian Motion and Lévy ProcessesConditionally Gaussian Distributions and Stochastic Volatility Models for the Discrete-time CaseMartingale Measures in the Stochastic Theory of ArbitrageChange of Measure in Option PricingConditionally Brownian and Lévy Processes. Stochastic Volatility ModelsA Wider View. Ambit Processes and Fields, and Volatility/Intermittency
Readership: Mathematical researchers, graduate students and practitioners interested in application of probabilistic theories & stochastic processes to economics & finance, and to turbulence.
Key Features:There is no other book on the market with such a specific and in-depth focus on these two main issues of change of time and change of probability law and Lévy measureIt is invaluable as a textbook for graduate-level courses and students, or a handy reference for researchers and practitioners in financial mathematics and econometricsProvides a strong and comprehensive account by drawing together sources from a wide range of books and research papers in the fields of theoretical and applied probability