Advanced. of Time. of Measure. Aarhus University, Denmark. Albert Shiryaev. Stek/ov Mathematical Institute and Moscow State University, Russia

Transcription

1 SHANGHAI TAIPEI Advanced Series on Statistical Science & Applied Probability Vol. I 3 Change and Change of Time of Measure Ole E. Barndorff-Nielsen Aarhus University, Denmark Albert Shiryaev Stek/ov Mathematical Institute and Moscow State University, Russia World Scientific NEW JERSEY LONDON SINGAPORE BEJJJNB HONG KONG CHENNAI

2 X Foreword Introduction v xi I. Random Change of Time Basic Definitions Some Properties of Change of Time Representations in the Weak Sense {X X ot), = in the Strong Sense (X XoT) = and the Semi-strong Sense (X a= X o T). I. Constructive Examples Representations in the Weak Sense (X '= X o T), Strong Sense o (X T) and the Semi-strong Sense (X a= X ot), II. The Case of Continuous Local Martingales and Processes of Bounded Variation Integral Representations and Change of Time in Stochas tic Integrals Integral Representations of Local Martingales in the Strong Sense Integral Representations of Local Martingales in a Semistrong Sense Stochastic Integrals Over the Stable Processes and Integral Representations Stochastic Integrals with Respect to Stable Processes and Change of Time Semimartingales: Basic Notions, Structures, Elements of vii

3 viii Stochastic Analysis Basic Definitions and Properties Canonical Representation. Triplets of Predictable Charac teristics Stochastic Integrals with Respect to a Brownian Motion, Square-integrable Martingales, and... Semimartingales Stochastic Differential Equations Stochastic Exponential and Stochastic Logarithm. Cumulant Processes Stochastic Exponential and Stochastic Logarithm Fourier Cumulant Processes Laplace Cumulant Processes Cumulant Processes of Stochastic Integral Transformation Xv = <p-x Processes with Independent Increments. Levy Processes Processes with Independent Increments and Semimartingales Processes with Stationary Independent Increments (Levy Processes) Some Properties of Sample Paths of Processes with Inde pendent Increments Some Properties of Sample Paths of Processes with Sta tionary Independent Increments (Levy Processes) Change of Measure. General Facts Basic Definitions. Density Process Discrete Version of Girsanov's Theorem Semimartingale Version of Girsanov's Theorem Esscher's Change of Measure Change of Measure in Models Based on Levy Processes Linear and Exponential Levy Models under Change of Measure On the Criteria of Local Absolute Continuity of Two Mea sures of Levy Processes 142

4 ix 7.3 On the Uniqueness of Locally Equivalent Martingale-type Measures for the Exponential Levy Models On the Construction of Martingale Measures with Minimal Entropy in the Exponential Levy Models Change of Time in Semimartingale Models and Models Based on Brownian Motion and Levy Processes Some General Facts about Change of Time for Semimar tingale Models Change of Time in Brownian Motion. Different Formu lations Change of Time Given by Subordinators. I. Some Ex amples Change of Time Given by Subordinators. II. Structure of the Triplets of Predictable Characteristics Conditionally Gaussian Distributions and Stochastic Volatility Models for the Discrete-time Case Deviation from the Gaussian Property of the Returns of the Prices Martingale Approach to the Study of the Returns of the Prices Conditionally Gaussian Models. I. Linear (AR, MA, ARMA) and Nonlinear (ARCH, GARCH) Models for Returns Conditionally Gaussian Models. II. IG- and GIGdistributions for the Square of Stochastic Volatility and GH-distributions for Returns Martingale Measures in the Stochastic Theory of Arbitrage Basic Notions and Summary of Results of the Theory of Arbitrage. I. Discrete Time Models Basic Notions and Summary of Results of the Theory of Arbitrage. II. Continuous-Time Models Arbitrage in a Model of Buying/Selling Assets with Trans action Costs Asymptotic Arbitrage: Some Problems Change of Measure in Option Pricing 225

5 X 11.1 Overview of the Pricing Formulae for European Options Overview of the Pricing Formulae for American Options Duality and Symmetry of the Semimartingale Models Call-Put Duality in Option Pricing. Levy Models Conditionally Brownian and Levy Processes. Stochastic Volatility Models From Black-Scholes Theory of Pricing of Derivatives to the Implied Volatility, Smile Effect and Stochastic Volatil ity Models Generalized Inverse Gaussian Subordinator and General ized Hyperbolic Levy Motion: Two Methods of Construc tion, Sample Path Properties Distributional and Sample-path Properties of the Levy Processes L(GIG) and L(GH) On Some Others Models of the Dynamics of Prices. Com parison of the Properties of Different Models 283 Afterword 289 Bibliography 291 Index 301