Change of Time and Change of Measure

by Barndorff-Nielsen, Ole E.; Shiryaev, Albert

• ISBN13: 9789814324472
• ISBN10: 9814324477
• Format: Hardcover
• Copyright: 2010-10-30
• Publisher: Textstream

Summary

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 a key to understanding the nature of various types of stochastic processes, and gives rise to interesting mathematical results and to 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, but also has a considerable role in insurance mathematics, in 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. It is invaluable as a textbook for graduate-level courses and students or a handy reference for researchers and practitioners in financial mathematics and econometrics.

Table of Contents

Foreword	p. v
Introduction	p. xi
Random Change of Time	p. 1
Basic Definitions	p. 1
Some Properties of Change of Time	p. 4
Representations in the Weak Sense (X $$$= X $$$ T), in the Strong Sense (X = X $$$ T) and the Semi-strong Sense (X $$$= X $$$ T). I. Constructive Examples	p. 8
Representations in the Weak Sense (X $$$= X $$$ T), Strong-Sense (X = X $$$ T) and the Semi-strong Sense (X $$$= X $$$ T). II. The Case of Continuous Local Martingales and Processes of Bounded Variation	p. 15
Integral Representations and Change of Time in Stochastic Integrals	p. 25
Integral Representations of Local Martingales in the Strong Sense	p. 25
Integral Representations of Local Martingales in a Semi-strong Sense	p. 33
Stochastic Integrals Over the Stable Processes and Integral Representations	p. 35
Stochastic Integrals with Respect to Stable Processes and Change of Time	p. 38
Semimartingales: Basic Notions, Structures, Elements of Stochastic Analysis	p. 41
Basic Definitions and Properties	p. 41
Canonical Representation. Triplets of Predictable Characteristics	p. 52
Stochastic Integrals with Respect to a Brownian Motion, Square-integrable Martingales, and Semimartingales	p. 56
Stochastic Differential Equations	p. 73
Stochastic Exponential and Stochastic Logarithm. Cumulant Processes	p. 91
Stochastic Exponential and Stochastic Logarithm	p. 91
Fourier Cumulant Processes	p. 96
Laplace Cumulant Processes	p. 99
Cumulant Processes of Stochastic Integral Transformation X¿ = ¿ X	p. 101
Processes with Independent Increments. Lévy Processes	p. 105
Processes with Independent Increments and Semimartingales	p. 105
Processes with Stationary Independent Increments (Lévy Processes)	p. 108
Some Properties of Sample Paths of Processes with Independent Increments	p. 113
Some Properties of Sample Paths of Processes with Stationary Independent Increments (Lévy Processes)	p. 117
Change of Measure. General Facts	p. 121
Basic Definitions. Density Process	p. 121
Discrete Version of Girsanov's Theorem	p. 123
Semimartingale Version of Girsanov's Theorem	p. 126
Esscher's Change of Measure	p. 132
Change of Measure in Models Based on Lévy Processes	p. 135
Linear and Exponential Lévy Models under Change of Measure	p. 135
On the Criteria of Local Absolute Continuity of Two Measures of Lévy Processes	p. 142
On the Uniqueness of Locally Equivalent Martingale-type Measures for the Exponential Lévy Models	p. 144
On the Construction of Martingale Measures with Minimal Entropy in the Exponential Lévy Models	p. 147
Change of Time in Semimartingale Models and Models Based on Brownian Motion and Lévy Processes	p. 151
Some General Facts about Change of Time for Semimartingale Models	p. 151
Change of Time in Brownian Motion. Different Formulations	p. 154
Change of Time Given by Subordinators. I Some Examples	p. 156
Change of Time Given by Subordinators. II. Structure of the Triplets of Predictable Characteristics	p. 158
Conditionally Gaussian Distributions and Stochastic Volatility Models for the Discrete-time Case	p. 163
Deviation from the Gaussian Property of the Returns of the Prices	p. 163
Martingale Approach to the Study of the Returns of the Prices	p. 166
Conditionally Gaussian Models. I. Linear (AR, MA, ARMA) and Nonlinear (ARCH, GARCH) Models for Returns	p. 171
Conditionally Gaussian Models. II. IG- and GIG-distributions for the Square of Stochastic Volatility and GH-distributions for Returns	p. 175
Martingale Measures in the Stochastic Theory of Arbitrage	p. 195
Basic Notions and Summary of Results of the Theory of Arbitrage. I. Discrete Time Models	p. 195
Basic Notions and Summary of Results of the Theory of Arbitrage. II. Continuous-Time Models	p. 207
Arbitrage in a Model of Buying/Selling Assets with Transaction Costs	p. 215
Asymptotic Arbitrage: Some Problems	p. 216
Change of Measure in Option Pricing	p. 225
Overview of the Pricing Formulae for European Options	p. 225
Overview of the Pricing Formulae for American Options	p. 240
Duality and Symmetry of the Semimartingale Models	p. 243
Call-Put Duality in Option Pricing. Lévy Models	p. 254
Conditionally Brownian and Lévy Processes. Stochastic Volatility Models	p. 259
From Black-Scholes Theory of Pricing of Derivatives to the Implied Volatility, Smile Effect and Stochastic Volatility Models	p. 259
Generalized Inverse Gaussian Subordinator and Generalized Hyperbolic Lévy Motion: Two Methods of Construction, Sample Path Properties	p. 270
Distributional and Sample-path Properties of the Lévy Processes L(GIG) and L(GH)	p. 275
On Some Others Models of the Dynamics of Prices. Comparison of the Properties of Different Models	p. 283
Afterword	p. 289
Bibliography	p. 291
Index	p. 301
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