扩散 马尔可夫过程和鞅 第1卷PDF|Epub|txt|kindle电子书版本网盘下载

扩散 马尔可夫过程和鞅 第1卷PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

CHAPTER Ⅰ．BROWNIAN MOTION1

1．INTRODUCTION1

1 What is Brownian motion,and why study it?1

2．Brownian motion as a martingale2

3．Brownian motion as a Gaussian process3

4．Brownian motion as a Markov process5

5 Brownian motion as a diffusion（and martingale）7

2．BASICS ABOUT BROWNIAN MOTION10

6 Existence and uniqueness of Brownian motion10

7．Skorokhod embedding13

8．Donsker's Invariance Principle16

9．Exponential martingales and first-passage distributions18

10．Some sample-path properties19

11．Quadratic variation21

12．The strong Markov property21

13．Reflection25

14．Reflecting Brownian motion and local time27

15．Kolmogorov's test31

16．Brownian exponential martingales and the Law of the Iterated Logarithm31

3．BROWNIAN MOTION IN HIGHER DIMENSIONS36

17．Some martingales for Brownian motion36

18．Recurrence and transience in higher dimensions38

19．Some applications of Brownian motion to complex analysis39

20．Windings of planar Brownian motion43

21．Multiple points,cone points,cut points45

22．Potential theory of Brownian motion in IRd（d≥3）46

23．Brownian motion and physical diffusion51

4．GAUSSIAN PROCESSES AND LEVY PROCESSES55

Gaussian processes55

24．Existence results for Gaussian processes55

25 Continuity results59

26．Isotropic random flows66

27．Dynkin's Isomorphism Theorem71

Lévy processes73

28．Lévy processes73

29．Fluctuation theory and Wiener-Hopf factorisation80

30．Local time of Levy processes82

CHAPTER Ⅱ．SOME CLASSICAL THEORY85

1．BASIC MEASURE THEORY85

Measurability and measure85

1．Measurable spaces;σ-algebras;π-systems;d-systems85

2．Measurable functions88

3．Monotone-Class Theorems90

4．Measures;the uniqueness lemma;almost everywhere;a.e．（μ，∑）91

5．Carathéodory's Extension Theorem93

6．Inner and outerμ-measures;completion94

Integration95

7．Definition of the integral∫fdμ95

8．Convergence theorems96

9．The Radon-Nikodym Theorem;absolute continuity;λ《μnotation;equivalent measures98

10．Inequalities;Lp and Lp spaces（p≥1）99

Product structures101

11．Productσ-algebras101

12．Product measure;Fubini's Theorem102

13．Exercises104

2．BASIC PROBABILITY THEORY108

Probability and expectation108

14．Probability triple;almost surely （a.s.）;a.s.（P），a.s.（P，F）108

15．lim sup En;First Borel-Cantelli Lemma109

16．Law of random variable;distribution function;joint law110

17．Expectation;E（X;F）110

18．Inequalities：Markov,Jensen,Schwarz,Tchebychev111

19．Modes of convergence of random variables113

Uniform integrability and L1 convergence114

20．Uniform integrability114

21．L1 convergence115

Independence116

22．Independence ofσ-algebras and of random variables116

23 Existence of families of independent variables118

24．Exercises119

3．STOCHASTIC PROCESSES119

The Daniell-Kolmogorov Theorem119

25．（ET，ET）;σ-algebras on function space;cylinders andσ-cylinders119

26．Infinite products of probability triples121

27．Stochastic process;sample function;law121

28．Canonical process122

29．Finite-dimensional distributions;sufficiency;compatibility123

30．The Daniell-Kolmogorov（DK）Theorem：compact metrizable'case124

31．The Daniell-Kolmogorov（DK）Theorem：general case126

32．Gaussian processes;pre-Brownian motion127

33．Pre-Poisson set functions128

Beyond the DK Theorem128

34．Limitations of the DK Theorem128

35．The role of outer measures129

36．Modifications;indistinguishability130

37．Direct const ruction of Poisson measures and subordinators,and of local time from the zero set;Azéma's martingale131

38．Exercises136

4．DISCRETE-PARAMETER MARTINGALE THEORY137

Conditional expectation137

30．Fundamental theorem and definition137

40．Notation;agreement with elementary usage138

41．Properties of conditional expectation：a list139

42．The role of versions;regular conditional probabilities and pdfs140

43．A counterexample141

44．A uniform-integrability property of conditional expectations （Discrete-parameter）martingales and supermartingales142

45．Filtration;filtered space;adapted process;natural filtration143

46 Martingale;supermartingale;submartingale144

47 Previsible process;gambling strategy;a fundamental principle144

48．Doob's Upcrossing Lemma145

49．Doob's Supermartingale-Convergence Theorem146

50．L1 convergence and the UI property147

51．The Lévy-Doob Downward Theorem148

52．Doob's Submartingale and Lp Inequalities150

53．Martingales in L2：orthogonality of increments152

54．Doob decomposition153

55．The〈M〉and［M］processes154

Stopping times,optional stopping and optional sampling155

56．Stopping time155

57．Optional-stopping theorems156

58．The pre-Tσ-algebraFT158

59．Optional sampling159

60．Exercises161

5．CONTINUOUS-PARAMETER SUPERMARTINGALES163

Regularisation：R-supermartingales163

61．Orientation163

62．Some real-variable results163

63．Filtrations;supermartingales;R-processes,R-supermartingales166

64．Some important examples167

65．Doob's Regularity Theorem：Part 1169

66．Partial augmentation171

67．Usual conditions;R-filtered space;usual augmentation;R-regularisation172

68．A necessary pause for thought174

69．Convergence theorems for R-supermartingales175

70．Inequalities and Lp convergence for R-submartingales177

71．Martingale proof of Wiener's Theorem;canonical Brownian motion178

72．Brownian motion relative to a filtered space180

Stopping times181

73．Stopping time T;pre-Tσ-algebra GT;progressive process181

74．First-entrance（début）times;hitting times;first-approach times：the easy cases183

75．Why‘completion'in the usual conditions has to be introduced184

76 Début and Section Theorems186

77．Optional Sampling for R-supermartingales under the usual conditions188

78 Two important results for Markov-process theory191

79．Exercises192

6．PROBABI LITY M EASU RES ON LUSIN SPACES200

‘Weak convergence'202

80 C（J）and Pr（J）when J is compact Hausdorff202

81．C（J）and Pr（J）when J is compact metrizable203

82．Polish and Lusin spaces205

83．The Cb（S）topology of Pr（S）when S is a Lusin space;Prohorov's Theorem207

84．Some useful convergence results211

85．Tightness in Pr（W）when W is the path-space W：=C（［0，∞）;IR）213

86．The Skorokhod representation of Ch（S）convergence on Pr（S）215

87．Weak convergence versus convergence of finite-dimensional distributions216

Regular conditional probabilities217

88．Some preliminaries217

89．The main existence theorem218

90．Canonical Brownian Motion CBM（IRN）;Markov property of Px laws220

91．Exercises222

CHAPTER Ⅲ．MARKOV PROCESSES227

1．TRANSITION FUNCTIONS AND RESOLVENTS227

1．What is a（continuous-time）Markov process?227

2．The finite-state-space Markov chain228

3．Transition functions and their resolvents231

4．Contraction semigroups on Banach spaces234

5．The Hille-Yosida Theorem237

2．FELLER-DYNKIN PROCESSES240

6．Feller-Dynkin（FD）semigroups240

7．The existence theorem：canonical FD processes243

8．Strong Markov property：preliminary version247

9．Strong Markov property：full version;Blumenthal's 0-1 Law249

10．Some fundamental martingales；Dynkin's formula252

11．Quasi-left-continuity255

12．Characteristic operator256

13．Feller-Dynkin diffusions258

14．Characterisation of continuous real Lévy processes261

15．Consolidation262

3．ADDITIVE FUNCTIONALS263

16．PCHAFs；λ-excessive functions；Brownian local time263

17．Proof of the Volkonskii-？ur-Meyer Theorem267

18．Killing269

19．The Feynmann-Kac formula272

20．A Ciesielski-Taylor Theorem275

21．Time-substitution277

22 Reflecting Brownian motion278

23．The Feller-McKean chain281

24．Elastic Brownian motion；the arcsine law282

4．APPROACH TO RAY PROCESSES：THE MARTIN BOUNDARY284

25．Ray processes and Markov chains284

26．Important example：birth process286

27．Excessive functions,the Martin kernel and Choquet theory288

28．The Martin compactification292

29．The Martin representation;Doob-Hunt explanation295

30．R.S.Martin's boundary297

31．Doob-Hunt theory for Brownian motion298

32．Ray processes and right processes302

5．RAY PROCESSES303

33．Orientation303

34．Ray resolvents304

35．The Ray-Knight compactification306

Ray's Theorem：analytical part309

36．From semigroup to resolvent309

37．Branch-points313

38．Choquet rep resentation of l-excessive probability measures315

Ray's Theorem：probabilistic part316

39．The Ray process associated with a given entrance law316

40．Strong Markov property of Ray processes318

41．The role of branch-points319

6．APPLICATIONS321

Martin boundary theory in retrospect321

42．From discrete to continuous time321

43．Proof of the Doob-Hunt Convergence Theorem323

44．The Choquet representation of ∏-excessive functions325

45．Doob h-transforms327

Time reversal and related topics328

46．Nagasawa's formula for chains328

47．Strong Markov property under time reversal330

48．Equilibrium charge331

49．BM（IR）and BES（3）：splitting times332

A first look at Markov-chain theory334

50．Chains as Ray processes334

51．Significance of qi337

52．Taboo probabilities;first-entrance decomposition337

53．The Q-matrix;DK conditions339

54．Local-character condition for Q340

55．Totally instantaneous Q-matrices342

56．Last exits343

57．Excursions from b345

58．Kingman's solution of the‘Markov characterization problem'347

59．Symmetrisable chains348

60．An open problem349

References for Volumes 1 and 2351

Index to Volumes 1 and 2375