Chapter
12. The strong Markov property
14. Reflecting Brownian motion and local time
16. Brownian exponential martingales and the Law of the
Iterated Logarithm
3. Brownian Motion in Higher Dimensions
17. Some martingales for Brownian motion
18. Recurrence and transience in higher dimensions
19. Some applications of Brownian motion to complex analysis
20. Windings of planar Brownian motion
21. Multiple points, cone points, cut points
22. Potential theory of Brownian motion in R[sup(d)] (d ≥ 3)
23. Brownian motion and physical diffusion
4. Gaussian Processes and Lévy Processes
24. Existence results for Gaussian processes
26. Isotropic random flows
27.
Dynkin's Isomorphism Theorem
29.
Fluctuation theory and Wiener-Hopf factorisation
30. Local time of Lévy processes
Chapter II. Some Classical
Theory
Measurability and measure
1. Measurable spaces; σ-algebras; π-systems;
d-systems
3. Monotone-Class Theorems
4. Measures; the uniqueness lemma; almost everywhere; a.e.(μ,
∑)
5. Carathéodory's
Extension Theorem
6. Inner and outer
μ-measures; completion
7. Definition of the integral ∫ f dμ
9. The Radon-Nikodým Theorem; absolute continuity; λ « μ
notation; equivalent measures
10. Inequalities; ℒ[sup(P)] and L[sup(P)] spaces (p ≥
1)
12. Product measure; Fubini's Theorem
2. Basic Probability Theory
Probability and expectation
14. Probability triple; almost surely (a.s.); a.s.(P), a.s.(P,ℱ)
15. lim sup E[sub(n)]; First Borel–Cantelli Lemma
16. Law of random variable; distribution function; joint law
18. Inequalities: Markov, Jensen, Schwarz, Tchebychev
19. Modes of convergence of random variables
Uniform
integrability and ℒ[sup(1)] convergence
20. Uniform integrability
21. ℒ[sup(1)]
convergence
22. Independence of
σ-algebras and of random variables
23. Existence of families of independent variables
The Daniell–Kolmogorov Theorem
25. (E[sup(T)], ℰ[sup(T)]); σ-algebras on function space; cylinders and
σ-cylinders
26. Infinite products of probability triples
27. Stochastic process; sample function; law
29. Finite-dimensional distributions; sufficiency; compatibility
30. The Daniell–Kolmogorov (DK) Theorem: 'compact metrizable' case
31. The Daniell–Kolmogorov (DK) Theorem: general case
32. Gaussian processes; pre-Brownian motion
33. Pre-Poisson set functions
34. Limitations of the DK Theorem
35. The role of outer measures
36. Modifications; indistinguishability
37. Direct construction of Poisson measures and subordinators, and of local time from the zero set; Azéma's
martingale
4. Discrete-Parameter Martingale Theory
39.
Fundamental theorem and definition
40. Notation; agreement with elementary usage
41. Properties of conditional expectation: a list
42. The role of versions; regular conditional probabilities and pdfs
44.
A uniform-integrability property of conditional expectations
(Discrete-parameter) martingales and supermartingales
45.
Filtration; filtered space; adapted process; natural filtration
46. Martingale; supermartingale; submartingale
47. Previsible process; gambling strategy; a fundamental principle
48. Doob's Upcrossing Lemma
49. Doob's Supermartingale-Convergence Theorem
50. ℒ[sup(1)] convergence and the UI property
51. The Lévy–Doob Downward Theorem
52. Doob's Submartingale and ℒ[sup(P)] Inequalities
53. Martingales in ℒ[sup(2)]; orthogonality of increments
55. The〈M〉and [M] processes
Stopping times, optional stopping and optional sampling
57. Optional-stopping theorems
58. The pre-T σ-algebra ℱ[sub(T)]
5. Continuous-Parameter Supermartingales
Regularisation: R-supermartingales
62. Some real-variable results
63. Filtrations; supermartingales; R-processes, R-supermartingales
64. Some important examples
65. Doob's Regularity Theorem: Part 1
67. Usual conditions; R-filtered space; usual augmentation;
R-regularisation
68. A necessary pause for thought
69. Convergence theorems for R-supermartingales
70. Inequalities and ℒ[sup(P)]
convergence for R-submartingales
71. Martingale proof of Wiener's Theorem; canonical
Brownian motion
72. Brownian motion relative to a filtered space
73. Stopping time T; pre-T σ-algebra G[sub(T)]; progressive
process
74. First-entrance
(début) times; hitting times; first-approach times: the easy cases
75. Why 'completion' in the usual conditions has to be introduced
76.
Début and Section Theorems
77. Optional Sampling for R-supermartingales under the
usual conditions
78. Two important results for Markov-process theory
6. Probability Measures on Lusin Spaces
80. C(J) and Pr(J)
when J is compact Hausdorff
81. C(J) and Pr(J) when J is compact metrizable
82. Polish and Lusin spaces
83. The C[sub(b)](S) topology of Pr(S) when S is a Lusin space; Prohorov's Theorem
84. Some useful convergence results
85. Tightness in Pr(W) when W is the path-space W:= C([0,
∞);R)
86. The Skorokhod representation of C[sub(b)](S) convergence on Pr(S)
87. Weak convergence versus convergence of finite-dimensional
distributions
Regular conditional probabilities
89. The main existence theorem
90. Canonical Brownian Motion CBM(R[sup(N)]); Markov property of P[sup(x)] laws
Chapter III. Markov
Processes
1. Transition Functions and Resolvents
1. What is a (continuous-time) Markov process?
2. The finite-state-space Markov chain
3. Transition functions and their resolvents
4. Contraction semigroups on Banach spaces
5. The Hille–Yosida Theorem
2. Feller–Dynkin Processes
6. Feller–Dynkin (FD) semigroups
7. The existence theorem: canonical FD processes
8. Strong Markov property: preliminary version
9. Strong Markov property: full version; Blumenthal's 0-1
Law
10. Some fundamental martingales; Dynkin's formula
11. Quasi-left-continuity
12. Characteristic operator
13. Feller–Dynkin diffusions
14. Characterisation of continuous real Lévy processes
16. PCHAFs; λ-excessive
functions; Brownian local time
17. Proof of the Volkonskii–Šur–Meyer Theorem
19. The Feynmann–Kac formula
20. A Ciesielski–Taylor Theorem
22. Reflecting Brownian motion
23. The Feller–McKean chain
24. Elastic Brownian motion; the arcsine law
4. Approach to Ray Processes: The Martin Boundary
25. Ray processes and Markov chains
26. Important example: birth process
27. Excessive functions, the Martin kernel and Choquet theory
28. The Martin compactification
29. The Martin representation; Doob–Hunt explanation
30. R. S. Martin's boundary
31. Doob–Hunt theory for Brownian motion
32. Ray processes and right processes
35. The Ray–Knight compactification
Ray's Theorem: analytical part
36. From semigroup to resolvent
38. Choquet representation of 1-excessive probability measures
Ray's Theorem: probabilistic part
39. The Ray process associated with a given entrance law
40. Strong Markov property of Ray processes
41. The role of branch-points
Martin boundary theory in retrospect
42. From discrete to continuous time
43. Proof of the Doob–Hunt Convergence Theorem
44. The Choquet representation of
Π-excessive functions
Time reversal and related topics
46. Nagasawa's formula for chains
47. Strong Markov property under time reversal
49. BM(R)
and BES(3): splitting times
A first look at Markov-chain theory
50. Chains as Ray processes
51. Significance of q[sub(i)]
52. Taboo probabilities; first-entrance decomposition
53. The Q-matrix; DK conditions
54. Local-character condition for Q
55. Totally instantaneous Q-matrices
58. Kingman's solution of the 'Markov characterization problem'
References for Volumes 1 and 2
Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
( Cambridge Mathematical Library )
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