General Theory of Markov Processes (eBook)

Front Cover 1
General Theory of Markov Processes 4
Copyright Page 5
Contents 10
Chapter I. Fundamental Hypotheses 14
1. Markov Property, Transition Functions and Entrance Laws 14
2. The First Regularity Hypothesis 20
3. The Natural Filtration 23
4. Excessive Functions and the Resolvent 27
5. The Optional and Predictable s-Algebras 32
6. The Strong Markov Property 38
7. The Second Fundamental Hypothesis 43
8. Right Processes 51
9. Existence Theorem for Ray Resolvents 55
10. Hitting Times and the Fine Topology 64
Chapter II. Transformations 74
11. The Lifetime Formalisms 75
12. Killing at a Terminal Time 78
13. Mappings of the State Space 88
14. Concatenation of Processes 90
15. Cartesian Products 97
16. Space-time Processes 99
17. Completion of a Resolvent 102
18. Right Process in the Ray Topology 107
19. Realizations of Right Processes 110
20. Résumé of Notation and Hypotheses 117
Chapter III. Homogeneity 119
21. Measurability and the Big Shift 119
22. Construction of Projections 123
23. Relations between the s-Algebras 128
24. Homogeneous Processes and Perfection 136
25. Co-optional and Coterminal Times 151
26. Measurability on the Future 154
Chapter IV. Random Measures 158
27. Random Measures and Increasing Processes 159
28. Integrability Conditions 160
29. Shifts of Random Measures 164
30. Kernels Associated with Random Measures 165
31. Dual Projections 170
32. Integral Measurability 176
33. Characterization by Potentials 179
34. Representation of Potentials 182
35. Homogeneous Random Measures 185
36. Potential Functions and Operators 188
37. Left Potential Functions 194
38. Generating Homogeneous Random Measures 197
Chapter V. Ray-Knight Methods 202
39. The Ray Space 203
40. The Entrance Space 208
41. Meager Sets and Predictable Functions 214
42. Left Limits and Predictable Projections 218
43. The Predictable Projection Kernel 220
44. Topological Characterizations of Projections 223
45. Accessibility 226
46. Left Limits in the Original Topology 231
47. Quasi-Left-Continuity 232
48. Natural Processes 235
49. Balayage of Functions 244
Chapter VI. Stochastic Calculus 246
50. Local Martingales over a Right Process 246
51. Decomposition Theorems 252
52. Semimartingales and Stochastic Integrals 262
53. Finite Lifetime Considerations 267
Chapter VII: Multiplicative Functionals 272
54. Multiplicative Functionals and Terminal Times 272
55. Exact Perfection of a Weak MF 276
56. Exactly Subordinate Semigroups 280
57. Decreasing MF’s 285
58. m-Additivity 288
59. Left MF’s and Exceptional Sets 291
60. Measurability of a MF 297
61. Subprocess Generated by a Decreasing MF 299
62. Subprocess Generated by a Supermartingale MF 303
Chapter VIII. Additive Functionals 315
63. Classification of Additive Functionals 316
64. Fine Support 317
65. Time Change by the Inverse of an AF 318
66. Absolute Continuity 322
67. Balayage of an AF 328
68. Local Times 337
69. Additive Functionals of Subprocesses 339
70. Relative Predictable Projections 342
71. Exponential Formulas 347
72. Two Motivating Examples 352
73. Lévy Systems 355
74. Excursions from a Homogeneous Set 363
75. Characteristic Measures of an AF 371
Appendices 377
A0. Monotone Class Theorems 377
A1. Universal Completion and Trace 380
A2. Radon Spaces 382
A3. Kernels 388
A4. Lebesgue-Stieltjes Integrals 392
A5. Sketch of the General Theory of Processes 398
A6. Relative Martingales and Projections 407
References 417
Notation Index 424
Subject Index 426
Pure and Applied Mathematics 434