# Change Of Time And Change Of Measure

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## Overview

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. 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.

## Product Details

ISBN-13: 9789814324472 World Scientific Publishing Company, Incorporated 11/09/2010 Advanced Series On Statistical Science And Applied Probability , #13 324 6.10(w) x 9.00(h) x 0.90(d)

Foreword v

Introduction xi

1 Random Change of Time 1

1.1 Basic Definitions 1

1.2 Some Properties of Change of Time 4

1.3 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 8

1.4 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 15

2 Integral Representations and Change of Time in Stochastic Integrals 25

2.1 Integral Representations of Local Martingales in the Strong Sense 25

2.2 Integral Representations of Local Martingales in a Semi-strong Sense 33

2.3 Stochastic Integrals Over the Stable Processes and Integral Representations 35

2.4 Stochastic Integrals with Respect to Stable Processes and Change of Time 38

3 Semimartingales: Basic Notions, Structures, Elements of Stochastic Analysis 41

3.1 Basic Definitions and Properties 41

3.2 Canonical Representation. Triplets of Predictable Characteristics 52

3.3 Stochastic Integrals with Respect to a Brownian Motion, Square-integrable Martingales, and Semimartingales 56

3.4 Stochastic Differential Equations 73

4 Stochastic Exponential and Stochastic Logarithm. Cumulant Processes 91

4.1 Stochastic Exponential and Stochastic Logarithm 91

4.2 Fourier Cumulant Processes 96

4.3 Laplace Cumulant Processes 99

4.4 Cumulant Processes of Stochastic Integral Transformation Xφ = φ X 101

5 Processes with Independent Increments. Lévy Processes 105

5.1 Processes with Independent Increments and Semimartingales 105

5.2 Processes with Stationary Independent Increments (Lévy Processes) 108

5.3 Some Properties of Sample Paths of Processes with Independent Increments 113

5.4 Some Properties of Sample Paths of Processes with Stationary Independent Increments (Lévy Processes) 117

6 Change of Measure. General Facts 121

6.1 Basic Definitions. Density Process 121

6.2 Discrete Version of Girsanov's Theorem 123

6.3 Semimartingale Version of Girsanov's Theorem 126

6.4 Esscher's Change of Measure 132

7 Change of Measure in Models Based on Lévy Processes 135

7.1 Linear and Exponential Lévy Models under Change of Measure 135

7.2 On the Criteria of Local Absolute Continuity of Two Measures of Lévy Processes 142

7.3 On the Uniqueness of Locally Equivalent Martingale-type Measures for the Exponential Lévy Models 144

7.4 On the Construction of Martingale Measures with Minimal Entropy in the Exponential Lévy Models 147

8 Change of Time in Semimartingale Models and Models Based on Brownian Motion and Lévy Processes 151

8.1 Some General Facts about Change of Time for Semimartingale Models 151

8.2 Change of Time in Brownian Motion. Different Formulations 154

8.3 Change of Time Given by Subordinators. I Some Examples 156

8.4 Change of Time Given by Subordinators. II. Structure of the Triplets of Predictable Characteristics 158

9 Conditionally Gaussian Distributions and Stochastic Volatility Models for the Discrete-time Case 163

9.1 Deviation from the Gaussian Property of the Returns of the Prices 163

9.2 Martingale Approach to the Study of the Returns of the Prices 166

9.3 Conditionally Gaussian Models. I. Linear (AR, MA, ARMA) and Nonlinear (ARCH, GARCH) Models for Returns 171

9.4 Conditionally Gaussian Models. II. IG- and GIG-distributions for the Square of Stochastic Volatility and GH-distributions for Returns 175

10 Martingale Measures in the Stochastic Theory of Arbitrage 195

10.1 Basic Notions and Summary of Results of the Theory of Arbitrage. I. Discrete Time Models 195

10.2 Basic Notions and Summary of Results of the Theory of Arbitrage. II. Continuous-Time Models 207

10.3 Arbitrage in a Model of Buying/Selling Assets with Transaction Costs 215

10.4 Asymptotic Arbitrage: Some Problems 216

11 Change of Measure in Option Pricing 225

11.1 Overview of the Pricing Formulae for European Options 225

11.2 Overview of the Pricing Formulae for American Options 240

11.3 Duality and Symmetry of the Semimartingale Models 243

11.4 Call-Put Duality in Option Pricing. Lévy Models 254

12 Conditionally Brownian and Lévy Processes. Stochastic Volatility Models 259

12.1 From Black-Scholes Theory of Pricing of Derivatives to the Implied Volatility, Smile Effect and Stochastic Volatility Models 259

12.2 Generalized Inverse Gaussian Subordinator and Generalized Hyperbolic Lévy Motion: Two Methods of Construction, Sample Path Properties 270

12.3 Distributional and Sample-path Properties of the Lévy Processes L(GIG) and L(GH) 275

12.4 On Some Others Models of the Dynamics of Prices. Comparison of the Properties of Different Models 283

Afterword 289

Bibliography 291

Index 301