Semimartingales and their Statistical Inference

Transcription

1 Semimartingales and their Statistical Inference B.L.S. Prakasa Rao Indian Statistical Institute New Delhi, India CHAPMAN & HALL/CRC Boca Raten London New York Washington, D.C.

2 Contents Preface xi 1 Semimartingales Introduction Stochastic Processes 2 Martingales Doob-Meyer Decomposition Stochastic Integration 27 Stochastic Integrals with Respect to a Wiener Process 27 Stochastic Integration with Respect to a Square Integrable Martingale 39 Quadratic Characteristic and Quadratic Variation Processes 42 Central Limit Theorem Local Martingales 49 Stochastic Integral with Respect to a Local Martingale Some Inequalities for Local Martingales 56 Strang Law of Large Numbers 60 A Martingale Conditional Law 63 Limit Theorems for Continuous Local Martingales 65 Some Additional Results on Stochastic Integrals with Respect to Square Integrable Local Martingales Semimartingales 71 Stochastic Integral with Respect to a Semimartingale 73 Product Formulae for Semimartingales 74 Generalized Ito-Ventzell Formula 75

3 Convergence of Quadratic Variation of Semimartingales 77 Yoerup's Theorem for Local Martingales 78 Stochastic Differential Equations 86 Random Measures 87 Stochastic Integral with Respect to the Measure \i v Decomposition of Local Martingales Using Stochastic Integrals Girsanov's Theorem 97 Girsanov's Theorem for Semimartingales 101 Girsanov's Theorem for Semimartingales (Multidimensional Version) 102 Gaussian Martingales Limit Theorems for Semimartingales 105 Stahle Convergence of Semimartingales Diffusion-Type Processes 115 Diffusion Processes 115 Eigen Functions and Martingales 119 Stochastic Modeling 122 Examples of Diffusion Processes 123 Diffusion-Type Processes Point Processes 130 Univariate Point Process (Simple) 130 Multivariate Point Process 137 Doubly Stochastic Poisson Process 137 Stochastic Time Change 140 References Exponential Families of Stochastic Processes Introduction Exponential Families of Semimartingales Stochastic Time Transformation 165 References Asymptotic Likelihood Theory Introduction 171 Different Types of Information and Their Relationships Examples Asymptotic Likelihood Theory for a Class of Exponential Families of Semimartingales 185

4 3.4 Asymptotic Likelihood Theory for General Processes Exercises 196 References Asymptotic Likelihood Theory for Diffusion Processes with Jumps Introduction 201 Diffusions with Jumps Absolute Continuity for Measures Generated by Diffusions with Jumps Score Vector and Information Matrix Asymptotic Likelihood Theory for Diffusion Processes with Jumps 215 Consistency 215 Limiting Distribution Asymptotic Likelihood Theory for a Special Class of Exponential Families Examples Exercises 234 References Quasi Likelihood and Semimartingales Quasi Likelihood and Discrete Time Processes Quasi Likelihood and Continuous Time Processes Quasi Likelihood and Special Semimartingale 243 Optimality 246 Asymptotic Properties 252 Existence and Consistency of the Quasi Likelihood Estimator 253 Asymptotic Normality of the Quasi Likelihood Estimator Quasi Likelihood and Partially Specified Counting Processes Examples Exercises 266 References Local Asymptotic Behavior of Semimartingale Experiments Local Asymptotic Mixed Normality 271 Regularity Conditions 274

5 6.2 Local Asymptotic Quadraticity 282 Limiting Distribution Local Asymptotic Infinite Divisibility 293 Regularity Conditions 295 A Stochastic Dominated Convergence Theorem Local Asymptotic Normality Multiplicative Models 309 Counting Processes with Multiplicative Intensity Exercises 322 References Likelihood and Asymptotic Efficiency Fully Specified Likelihood (Factorizable Models) 329 Local Asymptotic Normality Partially Specified Likelihood Partial Likelihood and Asymptotic Efficiency Partially Specified Likelihood and Asymptotic Efficiency (Counting Processes) 352 Improvement of Preliminary Estimators 372 References Inference for Counting Processes Introduction 381 Nonhomogeneous Poisson Processes 381 Processes of Poisson Type Parametric Inference 387 Estimation for Nonhomogeneous Poisson Process 387 Asymptotic Properties of an MLE 390 Limit Behavior of the Likelihood Ratio Process 391 Central Limit Theorem 393 M-Estimation for a Nonhomogeneous Poisson Process Consistency 404 Asymptotic Normality Semiparametric Inference 414 Consistency 420 Asymptotic Distribution Nonparametric Inference 424 Estimation by the Kernel Method (Nonhomogeneous Poisson Process) 424 Estimation by the Method of Sieves 441 Estimation by the Method of Penalty Functions 449

6 Estimation by the Method of Martingale Estimators 454 Maximum Likelihood Estimation Additive-Multiplicative Hazard Models 471 References Inference for Semimartingale Regression Models Estimation by the Quasi-Least-Squares Method 481 Consistency 484 Asymptotic Normality Estimation by the Maximum Likelihood Method 495 Estimation of Parameters when the Characteristics of the Noise are Known 497 Estimation of the Characteristics of the Noise Estimation by the Method of Sieves Nonlinear Semimartingale Regression Models 526 References Applications to Stochastic Modeling Introduction Applications to Engineering and Economic Systems Application to Modeling of Neuron Movement in a Nervous System 538 References 541 A Doleans Measure for Semimartingales and Burkholder's Inequality for Martingales 543 A.l Doleans Measure 543 A.2 Burkholder's Inequality for Martingales 544 B Interchanging Stochastic Integration and Ordinary Differentiation and Fubini-Type Theorem for Stochastic Integrals 545 B.l Interchanging Stochastic Integration and Ordinary Differentiation 545 B.2 Fubini-Type Theorem for Stochastic Integrals 550 B.3 Sufficient Conditions for the Differentiability of an Ito Stochastic Integral 550 C The Fundamental Identity of Sequential Analysis 553

7 D Stieltjes-Lebesgue Calculus 555 D.l Product Formula 558 D.2 Application of Product Formula 560 D.3 Exponential Formula 561 E A Useful Lemma 565 F Contiguity 569 References 570 G Notes 573 Index 579