Chapter Ⅰ．The General Theory of Stochastic Processes,Semimartingales and Stochastic Integrals

1．Stochastic Basis,Stopping Times,Optionalσ-Field,Martingales

1a．Stochastic Basis

1b．Stopping Times

1c．The Optional σ-Field

1d．The Localization Procedure

1e．Martingales

1f．The Discrete Case

2．Predictable σ-Field,Predictable Times

2a．The Predictable σ-Field

2b．Predictable Times

2c．Totally Inaccessible Stopping Times

2d．Predictable Projection

2e．The Discrete Case

3．Increasing Processes

3a．Basic Properties

3b．Doob-Meyer Decomposition and Compensators of Increasing Processes

3c．Lenglart Domination Property

3d．The Discrete Case

4．Semimartingales and Stochastic Integrals

4a．Locally Square-Integrable Martingales

4b．Decompositions of a Local Martingale

4c．Semimartingales

4d．Construction of the Stochastic Integral

4e．Quadratic Variation of a Semimartingale and Ito's Formula

4f．Doléans-Dade Exponential Formula

4g．The Discrete Case

Chapter Ⅱ．Characteristics of Semimartingales and Processes with Independent Increments

1．Random Measures

1a．General Random Measures

1b．Integer-Valued Random Measures

1c．A Fundamental Example：Poisson Measures

1d．Stochastic Integral with Respect to a Random Measure

2．Characteristics of Semimartingales

2a．Definition of the Characteristics

2b．Integrability and Characteristics

2c．A Canonical Representation for Semimartingales

2d．Characteristics and Exponential Formula

3．Some Examples

3a．The Discrete Case

3b．More on the Discrete Case

3c．The"One-Point"Point Process and Empirical Processes

4．Semimartingales with Independent Increments

4a．Wiener Processes

4b．Poisson Processes and Poisson Random Measures

4c．Processes with Independent Increments and Semimartingales

4d．Gaussian Martingales

5．Processes with Independent Increments Which Are Not Semimartingales

5a．The Results

5b．The Proofs

6．Processes with Conditionally Independent Increments

7．Progressive Conditional Continuous PIIs

8．Semimartingales,Stochastic Exponential and Stochastic Logarithm

8a．More About Stochastic Exponential and Stochastic Logarithm

8b．Multiplicative Decompositions and Exponentially Special Semimartingales

Chapter Ⅲ．Martingale Problems and Changes of Measures

1．Martingale Problems and Point Processes

1a．General Martingale Problems

1b．Martingale Problems and Random Measures

1c．Point Processes and Multivariate Point Processes

2．Martingale Problems and Semimartingales

2a．Formulation of the Problem

2b．Example：Processes with Independent Increments

2c．Diflusion Processes and Diffusion Processes with Jumps

2d．Local Uniqueness

3．Absolutely Continuous Changes of Measures

3a．The Density Process

3b．Girsanov's Theorem for Local Martingales

3c．Girsanoy's Theorem for Random Measures

3d．Girsanov's Theorem for Semimartingales

3e．The Discrete Case

4．Representation Theorem for Martingales

4a．Stochastic Integrals with Respect to a Multi-Dimensional Continuous Local Martingale

4b．Projection of a Local Martingale on a Random Measure

4c．The Representation Property

4d．The Fundamental Representation Theorem

5．Absolutely Continuous Change of Measures：Explicit Computation of the Density Process

5a．All P-Martingales Have the Representation Property Relative to X

5b．P′Has the Local Uniqueness Property

5c．Examples

6．Integrals of Vector-Valued Processes and σ-martingales

6a．Stochastic Integrals with Respect to a Multi-Dimensional Locally Square-integrable Martingale

6b．Integrals with Respect to a Multi-Dimensional Process of Locally Finite Variation

6c．Stochastic Integrals with Respect to a Multi-Dimensional Semimartingale

6d．Stochastic Integrals：A Predictable Criterion

6e．∑-localization and σ-martingales

7．Laplace Cumulant Processes and Esscher's Change of Measures

7a．Laplace Cumulant Processes of Exponentially Special Semimartingales

7b．Esscher Change of Measure

Chapter Ⅳ．Hellinger Processes,Absolute Continuity and Singularity of Measures

1．Hellinger Integrals and Hellinger Processes

1a．Kakutani-Hellinger Distance and Hellinger Integrals

1b．Hellinger Processes

1c．Computation of Hellinger Processes in Terms of the Density Processes

1d．Some Other Processes of Interest

1e．The Discrete Case

2．Predictable Criteria for Absolute Continuity and Singularity

2a．Statement of the Results

2b．The Proofs

2c．The Discrete Case

3．Hellinger Processes for Solutions of Martingale Problems

3a．The General Setting

3b．The Case Where P and P′Are Dominated by a Measure Having the Martingale Representation Property

3c．The Case Where Local Uniqueness Holds

4．Examples

4a．Point Processes and Multivariate Point Processes

4b．Generalized Diffusion Processes

4c．Processes with Independent Increments

Chapter Ⅴ．Contiguity,Entire Separation,Convergence in Variation

1．Contiguity and Entire Separation

1a．General Facts

1b．Contiguity and Filtrations

2．Predictable Criteria for Contiguity and Entire Separation

2a．Statements of the Results

2b．The Proofs

2c．The Discrete Case

3．Examples

3a．Point Processes

3b．Generalized Diffusion Processes

3c．Processes with Independent Increments

4．Variation Metric

4a．Variation Metric and Hellinger Integrals

4b．Variation Metric and Hellinger Processes

4c．Examples：Point Processes and Multivariate Point Processes

4d．Example：Generalized Diffusion Processes

Chapter Ⅵ．Skorokhod Topology and Convergence of Processes

1．The Skorokhod Topology

1a．Introduction and Notation

1b．The Skorokhod Topology：Definition and Main Results

1c．Proof of Theorem 1.14

2．Continuity for the Skorokhod Topology

2a．Continuity Properties of some Functions

2b．Increasing Functions and the Skorokhod Topology

3．Weak Convergence

3a．Weak Convergence of Probability Measures

3b．Application to Càdlàg Processes

4．Criteria for Tightness：The Quasi-Left Continuous Case

4a．Aldous'Criterion for Tightness

4b．Application to Martingales and Semimartingales

5．Criteria for Tightness：The General Case

5a．Criteria for Semimartingales

5b．An Auxiliary Result

5c．Proof of Theorem 5.17

6．Convergence,Quadratic Variation,Stochastic Integrals

6a．The P-UT Condition

6b．Tightness and the P-UT Property

6c．Convergence of Stochastic Integrals and Quadratic Variation

6d．Some Additional Results

Chapter Ⅶ．Convergence of Processes with Independent Increments

1．Introduction to Functional Limit Theorems

2．Finite-Dimensional Convergence

2a．Convergence of Infinitely Divisible Distributions

2b．Some Lemmas on Characteristic Functions

2c．Convergence of Rowwise Independent Triangular Arrays

2d．Finite-Dimensional Convergence of PII-Semimartingales to a PII Without Fixed Time of Discontinuity

3．Functional Convergence and Characteristics

3a．The Results

3b．Sufficient Condition for Convergence Under2.48

3c．Necessary Condition for Convergence

3d．Sufficient Condition for Convergence

4．More on the General Case

4a．Convergence ofNon-Infinitesimal Rowwise Independent　Arrays

4b．Finite-Dimensional Convergence for General PII

4c．Another Necessary and Sufficient Condition for Functional Convergence

5．The Central Limit Theorem

5a．The Lindeberg-Feller Theorem

5b．Zolotarev's Type Theorems

5c．Finite-Dimensional Convergence of PII's to a Gaussian Martingale

5d．Functional Convergence of PII's to a Gaussian Martingale

Chapter Ⅷ．Convergence to a Process with Independent Increments

1．Finite-Dimensional Convergence,a General Theorem

1a．Description of the Setting for This Chapter

1b．The Basic Theorem

1c．Remarks and Comments

2．Convergence to a PII Without Fixed Time of Discontinuity

2a．Finite-Dimensional Convergence

2b．Functional Convergence

2c．Application to Triangular Arrays

2d．Other Conditions for Convergence

3．Applications

3a．Central Limit Theorem：Necessary and Sufficient Conditions

3b．Central Limit Theorem：The Martingale Case

3c．Central Limit Theorem for Triangular Arrays

3d．Convergence of Point Processes

3e．Normed Sums of I.I.D.Semimartingales

3f．Limit Theorems for Functionals of Markov Processes

3g．Limit Theorems for Stationary Processes

4．Convergence to a General Process with Independent Increments

4a．Proof of Theorem 4.1 When the Characteristic Function of Xt Vanishes Almost Nowhere

4b．Convergence of Point Processes

4c．Convergence to a Gaussian Martingale

5．Convergence to a Mixture of PII's,Stable Convergence and Mixing Convergence

5a．Convergence to a Mixture of PII's

5b．More on the Convergence to a Mixture of PII's

5c．Stable Convergence

5d．Mixing Convergence

5e．Application to Stationary Processes

Chapter Ⅸ．Convergence to a Semimartingale

1．Limits of Martingales

1a．The Bounded Case

1b．The Unbounded Case

2．Identification of the Limit

2a．Introductory Remarks

2b．Identification of the Limit：The Main Result

2c．Identification of the Limit Via Convergence of the Characteristics

2d．Application：Existence of Solutions to Some Martingale Problems

3．Limit Theorems for Semimartingales

3a．Tightness of the Sequence(Xn)

3b．Limit Theorems：The Bounded Case

3c．Limit Theorems：The Locally Bounded Case

4．Applications

4a．Convergence of Diffusion Processes with Jumps

4b．Convergence of Step Markov Processes to Diffusions

4c．Empirical Distributions and Brownian Bridge

4d．Convergence to a Continuous Semimartingale：Necessary and Sufficient Conditions

5．Convergence of Stochastic Integrals

5a．Characteristics of Stochastic Integrals

5b．Statement of the Results

5c．The Proofs

6．Stability for Stochastic Differential Equation

6a．Auxiliary Results

6b．Stochastic Differential Equations

6c．Stability

7．Stable Convergence to a Progressive Conditional Continuous PII

7a．A General Result

7b．Convergence of Discretized Processes

Chapter Ⅹ．Limit Theorems,Density Processes and Contiguity

1．Convergence of the Density Processes to a Continuous Process

1a．Introduction,Statement of the Main Results

1b．An Auxiliary Computation

1c．Proofs of Theorems 1.12 and 1.16

1d．Convergence to the Exponential of a Continuous Martingale

1e．Convergencein Terms of Hellinger Processes

2．Convergence of the Log-Likelihood to a Process with Independent Increments

2a．Introduction Statement of the Results

2b．The Proof of Theorem 2.12

2c．Example：Point Processes

3．The Statistical Invariance Principle

3a．General Results

3b．Convergence to a Gaussian Martingale

Bibliographical Comments

References

Index of Symbols

Index of Terminology

Index of Topics

Index of Conditions for Limit Theorems