<p></p><p>随机分析的经典教材。</p><p>djvu格式，阅读软件网上可以下载。非常清晰。</p><p>Limit_Theorems_for_Stochastic_Processes.djvu</p><p>  </p><p>Description:<br/>Initially the theory of convergence in law of stochastic processes was developed quite independently from the theory of martingales, semimartingales and stochastic integrals. Apart from a few exceptions essentially concerning diffusion processes, it is only recently that the relation between the two theories has been thoroughly studied. The authors of this Grundlehren volume, two of the international leaders in the field, propose a systematic exposition of convergence in law for stochastic processes, from the point of view of semimartingale theory, with emphasis on results that are useful for mathematical theory and mathematical statistics. This leads them to develop in detail some particularly useful parts of the general theory of stochastic processes, such as martingale problems, and absolute continuity or contiguity results. The book contains an introduction to the theory of martingales and semimartingales, random measures stochastic integrales, Skorokhod topology, etc., as well as a large number of results which have never appeared in book form, and some entirely new results. The second edition contains some additions to the text and references. Some parts are completely rewritten.<br/><br/>Table of Contents <br/>Chapter I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals 1 <br/>1. Stochastic Basis, Stopping Times, Optional a-Field, Martingales 1 <br/>§la. Stochastic Basis 2 <br/>§lb. Stopping Times 4 <br/>§lc. The Optional a-Field 5 <br/>§ld. The Localization Procedure 8 <br/>§le. Martingales 10 <br/>§lf. The Discrete Case 13 <br/>2. Predictable a-Field, Predictable Times 16 <br/>§2a. The Predictable a-Field 16 <br/>§2b. Predictable Times 17 <br/>§2c. Totally Inaccessible Stopping Times 20 <br/>§2d. Predictable Projection 22 <br/>§2e. The Discrete Case 25 <br/>3. Increasing Processes 27 <br/>§3a. Basic Properties 27 <br/>§3b. Doob-Meyer Decomposition and Compensators of Increasing Processes 32 <br/>§3c. Lenglart Domination Property 35 <br/>§3d. The Discrete Case 36 <br/>4. Semimartingales and Stochastic Integrals 38 <br/>§4a. Locally Square-Integrable Martingales 38 <br/>§4b. Decompositions of a Local Martingale 40 <br/>§4c. Semimartingales 43 <br/>§4d. Construction of the Stochastic Integral 46 <br/>§4e. Quadratic Variation of a Semimartingale and Ito's Formula .... 51 <br/>§4f. Doleans-Dade Exponential Formula 58 <br/>§4g. The Discrete Case 62 <br/>Chapter II. Characteristics of Semimartingales and Processes with Independent Increments 64 <br/>1. Random Measures 64 <br/>§la. General Random Measures 65 <br/>§lb. Integer-Valued Random Measures 68 <br/>§lc. A Fundamental Example: Poisson Measures 70 <br/>§ld. Stochastic Integral with Respect to a Random Measure 71 <br/>2. Characteristics of Semimartingales 75 <br/>§2a. Definition of the Characteristics 75 <br/>§2b. Integrability and Characteristics 81 <br/>§2c. A Canonical Representation for Semimartingales 84 <br/>§2d. Characteristics and Exponential Formula 85 <br/>3. Some Examples 91 <br/>§3a. The Discrete Case 91 <br/>§3b. More on the Discrete Case 93 <br/>§3c. The "One-Point" Point Process and Empirical Processes 97 <br/>4. Semimartingales with Independent Increments 101 <br/>§4a. Wiener Processes 102 <br/>§4b. Poisson Processes and Poisson Random Measures 103 <br/>§4c. Processes with Independent Increments and Semimartingales . . 106 <br/>§4d. Gaussian Martingales Ill <br/>5. Processes with Independent Increments Which Are Not Semimartingales 114 <br/>§5a. The Results 114 <br/>§5b. The Proofs 116 <br/>6. Processes with Conditionally Independent Increments 124 <br/>7. Progressive Conditional Continuous PIIs 128 <br/>8. Semimartingales, Stochastic Exponential and Stochastic Logarithm . . 134 <br/>§8a. More About Stochastic Exponential and Stochastic Logarithm . . 134 <br/>§8b. Multiplicative Decompositions and Exponentially Special Semimartingales 138 <br/>Chapter III. Martingale Problems and Changes of Measures 142 <br/>1. Martingale Problems and Point Processes 143 <br/>§la. General Martingale Problems 143 <br/>§lb. Martingale Problems and Random Measures 144 <br/>§lc. Point Processes and Multivariate Point Processes 146 <br/>2. Martingale Problems and Semimartingales 151 <br/>§2a. Formulation of the Problem 152 <br/>§2b. Example: Processes with Independent Increments 154 <br/>§2c. Diffusion Processes and Diffusion Processes with Jumps 155 <br/>§2d. Local Uniqueness 159 <br/>3. Absolutely Continuous Changes of Measures 165 <br/>§3a. The Density Process 165 <br/>§3b. Girsanov's Theorem for Local Martingales 168 <br/>§3c. Girsanoy's Theorem for Random Measures 170 <br/>§3d. Girsanov's Theorem for Semimartingales 172 <br/>§3e. The Discrete Case 177 <br/>4. Representation Theorem for Martingales 179 <br/>§4a. Stochastic Integrals with Respect to a Multi-Dimensional Continuous Local Martingale 179 <br/>§4b. Projection of a Local Martingale on a Random Measure 182 <br/>§4c. The Representation Property 185 <br/>§4d. The Fundamental Representation Theorem 187 <br/>5. Absolutely Continuous Change of Measures: Explicit Computation of the Density Process 191 <br/>§5a. All P-Martingales Have the Representation Property Relative to X 192 <br/>§5b. Pf Has the Local Uniqueness Property 196 <br/>§5c. Examples 200 <br/>6. Integrals of Vector-Valued Processes and a-martingales 203 <br/>§6a. Stochastic Integrals with Respect to a Multi-Dimensional Locally Square-integrable Martingale 204 <br/>§6b. Integrals with Respect to a Multi-Dimensional Process of Locally Finite Variation 206 <br/>§6c. Stochastic Integrals with Respect to a Multi-Dimensional Semimartingale 207 <br/>§6d. Stochastic Integrals: A Predictable Criterion 212 <br/>§6e. ^-localization and a-martingales 214 <br/>7. Laplace Cumulant Processes and Esscher's Change of Measures .... 219 <br/>§7a. Laplace Cumulant Processes of Exponentially Special Semimartingales 219 <br/>§7b. Esscher Change of Measure 222 <br/>Chapter IV. Hellinger Processes, Absolute Continuity and Singularity of Measures 227 <br/>1. Hellinger Integrals and Hellinger Processes 228 <br/>§la. Kakutani-Hellinger Distance and Hellinger Integrals 228 <br/>§lb. Hellinger Processes 230 <br/>§lc. Computation of Hellinger Processes in Terms of the Density Processes 234 <br/>§ld. Some Other Processes of Interest 237 <br/>§le. The Discrete Case 242 <br/>2. Predictable Criteria for Absolute Continuity and Singularity 245 <br/>§2a. Statement of the Results 245 <br/>§2b. The Proofs 248 <br/>§2c. The Discrete Case 252 <br/>3. Hellinger Processes for Solutions of Martingale Problems 254 <br/>§3a. The General Setting 255 <br/>§3b. The Case Where P and P' Are Dominated by a Measure Having the Martingale Representation Property 257 <br/>§3c. The Case Where Local Uniqueness Holds 266 <br/>4. Examples 272 <br/>§4a. Point Processes and Multivariate Point Processes 272 <br/>§4b. Generalized Diffusion Processes 275 <br/>§4c. Processes with Independent Increments 277 <br/>Chapter V. Contiguity, Entire Separation, Convergence in Variation . . . 284 <br/>1. Contiguity and Entire Separation 284 <br/>§ 1 a. General Facts 284 <br/>§lb. Contiguity and Filiations 290 <br/>2. Predictable Criteria for Contiguity and Entire Separation 291 <br/>§2a. Statements of the Results 291 <br/>§2b. The Proofs 294 <br/>§2c. The Discrete Case 301 <br/>3. Examples 304 <br/>§3a. Point Processes 304 <br/>§3b. Generalized Diffusion Processes 305 <br/>§3c. Processes with Independent Increments 306 <br/>4. Variation Metric 309 <br/>§4a. Variation Metric and Hellinger Integrals 310 <br/>§4b. Variation Metric and Hellinger Processes 312 <br/>Table of Contents XVII <br/>§4c. Examples: Point Processes and Multivariate Point Processes ... 318 <br/>§4d. Example: Generalized Diffusion Processes 322 <br/>Chapter VI. Skorokhod Topology and Convergence of Processes 324 <br/>1. The Skorokhod Topology 325 <br/>§la. Introduction and Notation 325 <br/>§lb. The Skorokhod Topology: Definition and Main Results 327 <br/>§lc. Proof of Theorem 1.14 329 <br/>2. Continuity for the Skorokhod Topology 337 <br/>§2a. Continuity Properties of some Functions 337 <br/>§2b. Increasing Functions and the Skorokhod Topology 342 <br/>3. Weak Convergence 347 <br/>§3a. Weak Convergence of Probability Measures 347 <br/>§3b. Application to Cadlag Processes 348 <br/>4. Criteria for Tightness: The Quasi-Left Continuous Case 355 <br/>§4a. Aldous' Criterion for Tightness 356 <br/>§4b. Application to Martingales and Semimartingales 358 <br/>5. Criteria for Tightness: The General Case 362 <br/>§5a. Criteria for Semimartingales 362 <br/>§5b. An Auxiliary Result 365 <br/>§5c. Proof of Theorem 5.17 367 <br/>6. Convergence, Quadratic Variation, Stochastic Integrals 376 <br/>§6a. The P-UT Condition 377 <br/>§6b. Tightness and the P-UT Property 382 <br/>§6c. Convergence of Stochastic Integrals and Quadratic Variation ... 382 <br/>§6d. Some Additional Results 386 <br/>Chapter VII. Convergence of Processes with Independent Increments . . 389 <br/>1. Introduction to Functional Limit Theorems 390 <br/>2. Finite-Dimensional Convergence 394 <br/>§2a. Convergence of Infinitely Divisible Distributions 394 <br/>§2b. Some Lemmas on Characteristic Functions 398 <br/>§2c. Convergence of Rowwise Independent Triangular Arrays 402 <br/>§2d. Finite-Dimensional Convergence of Pll-Semimartingales to a PII Without Fixed Time of Discontinuity 408 <br/>3. Functional Convergence and Characteristics 413 <br/>§3a. The Results 414 <br/>§3b. Sufficient Condition for Convergence Under 2.48 418 <br/>§3c. Necessary Condition for Convergence 418 <br/>§3d. Sufficient Condition for Convergence 424 <br/>4. More on the General Case 428 <br/>§4a. Convergence of Non-Infinitesimal Rowwise Independent Arrays 428 <br/>§4b. Finite-Dimensional Convergence for General PII 436 <br/>§4c. Another Necessary and Sufficient Condition for Functional <br/>Convergence 439 <br/>5. The Central Limit Theorem 444 <br/>§5a. The Lindeberg-Feller Theorem 445 <br/>§5b. Zolotarev's Type Theorems 446 <br/>§5c. Finite-Dimensional Convergence of PIP s to a Gaussian Martingale 450 <br/>§5d. Functional Convergence of PIPs to a Gaussian Martingale 452 <br/>Chapter VIII. Convergence to a Process with Independent Increments . . 456 <br/>1. Finite-Dimensional Convergence, a General Theorem 456 <br/>§la. Description of the Setting for This Chapter 456 <br/>§lb. The Basic Theorem 457 <br/>§lc. Remarks and Comments 459 <br/>2. Convergence to a PII Without Fixed Time of Discontinuity 460 <br/>§2a. Finite-Dimensional Convergence 461 <br/>§2b. Functional Convergence 464 <br/>§2c. Application to Triangular Arrays 465 <br/>§2d. Other Conditions for Convergence 467 <br/>3. Applications 469 <br/>§3a. Central Limit Theorem: Necessary and Sufficient Conditions . . . 470 <br/>§3b. Central Limit Theorem: The Martingale Case 473 <br/>§3c. Central Limit Theorem for Triangular Arrays 477 <br/>§3d. Convergence of Point Processes 478 <br/>§3e. Normed Sums of I.I.D. Semimartingales 481 <br/>§3f. Limit Theorems for Functionals of Markov Processes 486 <br/>§3g. Limit Theorems for Stationary Processes 489 <br/>4. Convergence to a General Process with Independent Increments .... 499 <br/>§4a. Proof of Theorem 4.1 When the Characteristic Function of Xt <br/>Vanishes Almost Nowhere 501 <br/>§4b. Convergence of Point Processes 503 <br/>§4c. Convergence to a Gaussian Martingale 504 <br/>5. Convergence to a Mixture of PII's, Stable Convergence and Mixing Convergence 506 <br/>§5a. Convergence to a Mixture of PII's 506 <br/>§5b. More on the Convergence to a Mixture of PII's 510 <br/>§5c. Stable Convergence 512 <br/>§5d. Mixing Convergence 518 <br/>§5e. Application to Stationary Processes 519 <br/>Chapter IX. Convergence to a Semimartingale 521 <br/>1. Limits of Martingales 521 <br/>§la. The Bounded Case 522 <br/>§lb. The Unbounded Case 524 <br/>2. Identification of the Limit 527 <br/>§2a. Introductory Remarks 527 <br/>§2b. Identification of the Limit: The Main Result 530 <br/>§2c. Identification of the Limit Via Convergence of the Characteristics 533 <br/>§2d. Application: Existence of Solutions to Some Martingale Problems 535 <br/>3. Limit Theorems for Semimartingales 540 <br/>§3a. Tightness of the Sequence (Xn) 541 <br/>§3b. Limit Theorems: The Bounded Case 546 <br/>§3c. Limit Theorems: The Locally Bounded Case 550 <br/>4. Applications 554 <br/>§4a. Convergence of Diffusion Processes with Jumps 554 <br/>§4b. Convergence of Step Markov Processes to Diffusions 557 <br/>§4c. Empirical Distributions and Brownian Bridge 560 <br/>§4d. Convergence to a Continuous Semimartingale: Necessary and Sufficient Conditions 561 <br/>5. Convergence of Stochastic Integrals 564 <br/>§5a. Characteristics of Stochastic Integrals 564 <br/>§5b. Statement of the Results 567 <br/>§5c. The Proofs 570 <br/>6. Stability for Stochastic Differential Equation 575 <br/>§6a. Auxiliary Results 576 <br/>§6b. Stochastic Differential Equations 577 <br/>§6c. Stability 578 <br/>7. Stable Convergence to a Progressive Conditional Continuous PII. . . . 583 <br/>§7a. A General Result 583 <br/>§7b. Convergence of Discretized Processes 589 <br/>Chapter X. Limit Theorems, Density Processes and Contiguity 592 <br/>1. Convergence of the Density Processes to a Continuous Process 593 <br/>§la. Introduction, Statement of the Main Results 593 <br/>§lb. An Auxiliary Computation 597 <br/>§lc. Proofs of Theorems 1.12 and 1.16 603 <br/>§ 1 d. Convergence to the Exponential of a Continuous Martingale . . . 606 <br/>§le. Convergence in Terms of Hellinger Processes 609 <br/>2. Convergence of the Log-Likelihood to a Process <br/>with Independent Increments 612 <br/>§2a. Introduction, Statement of the Results 612 <br/>§2b. The Proof of Theorem 2.12 615 <br/>§2c. Example: Point Processes 619 <br/>3. The Statistical Invariance Principle 620 <br/>§3a. General Results 621 <br/>§3b. Convergence to a Gaussian Martingale 623 <br/>Bibliographical Comments 629 <br/>References 641 <br/>Index of Symbols 653 <br/>Index of Terminology 655 <br/>Index of Topics 659 <br/>Index of Conditions for Limit Theorems 661 </p><br/>