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Format: 1 online resource (xx, 386 pages) : , digital, PDF file(s).

Edition: 2nd edition.

ISBN: 1-107-71349-8 , 1-107-71070-7 , 1-107-59012-4

Series Statement: Cambridge mathematical library

Content: Now available in paperback, this celebrated book has been prepared with readers' needs in mind, remaining a systematic guide to a large part of the modern theory of Probability, whilst retaining its vitality. The authors' aim is to present the subject of Brownian motion not as a dry part of mathematical analysis, but to convey its real meaning and fascination. The opening, heuristic chapter does just this, and it is followed by a comprehensive and self-contained account of the foundations of theory of stochastic processes. Chapter 3 is a lively and readable account of the theory of Markov processes. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Note: Title from publisher's bibliographic system (viewed on 05 Oct 2015). , Cover -- Half Title -- Title Page -- Copyright -- Dedication -- Contents -- Some Frequently Used Notation -- Chapter I. Brownian Motion -- 1. Introduction -- 1. What is Brownian motion, and why study it? -- 2. Brownian motion as a martingale -- 3. Brownian motion as a Gaussian process -- 4. Brownian motion as a Markov process -- 5. Brownian motion as a diffusion (and martingale) -- 2. Basics About Brownian Motion -- 6. Existence and uniqueness of Brownian motion -- 7. Skorokhod embedding -- 8. Donsker's Invariance Principle -- 9. Exponential martingales and first-passage distributions -- 10. Some sample-path properties -- 11. Quadratic variation -- 12. The strong Markov property -- 13. Reflection -- 14. Reflecting Brownian motion and local time -- 15. Kolmogorov's test -- 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 -- Gaussian processes -- 24. Existence results for Gaussian processes -- 25. Continuity results -- 26. Isotropic random flows -- 27. Dynkin's Isomorphism Theorem -- Lévy processes -- 28. Lévy processes -- 29. Fluctuation theory and Wiener-Hopf factorisation -- 30. Local time of Lévy processes -- Chapter II. Some Classical Theory -- 1. Basic Measure Theory -- Measurability and measure -- 1. Measurable spaces -- σ-algebras -- π-systems -- d-systems -- 2. Measurable functions -- 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 -- Integration -- 7. Definition of the integral ∫ f dμ -- 8. Convergence theorems -- 9. The Radon-Nikodým Theorem -- absolute continuity -- λ « μ notation -- equivalent measures -- 10. Inequalities -- ℒ[sup(P)] and L[sup(P)] spaces (p ≥ 1) -- Product structures -- 11. Product σ-algebras -- 12. Product measure -- Fubini's Theorem -- 13. Exercises -- 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 -- 17. Expectation -- E(X -- F) -- 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 -- Independence -- 22. Independence of σ-algebras and of random variables -- 23. Existence of families of independent variables -- 24. Exercises -- 3. Stochastic Processes -- 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 -- 28. Canonical process -- 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 -- Beyond the DK Theorem -- 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 -- 38. Exercises -- 4. Discrete-Parameter Martingale Theory -- Conditional expectation -- 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 -- 43. A counterexample -- 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 -- 54. Doob decomposition -- 55. The〈M〉and [M] processes -- Stopping times, optional stopping and optional sampling -- 56. Stopping time -- 57. Optional-stopping theorems -- 58. The pre-T σ-algebra ℱ[sub(T)] -- 59. Optional sampling -- 60. Exercises -- 5. Continuous-Parameter Supermartingales -- Regularisation: R-supermartingales -- 61. Orientation -- 62. Some real-variable results -- 63. Filtrations -- supermartingales -- R-processes, R-supermartingales -- 64. Some important examples -- 65. Doob's Regularity Theorem: Part 1 -- 66. Partial augmentation -- 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 -- Stopping times -- 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 -- 79. Exercises -- 6. Probability Measures on Lusin Spaces -- 'Weak convergence' -- 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 -- 88. Some preliminaries -- 89. The main existence theorem -- 90. Canonical Brownian Motion CBM(R[sup(N)]) -- Markov property of P[sup(x)] laws -- 91. Exercises -- 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 -- 15. Consolidation -- 3. Additive Functionals -- 16. PCHAFs -- λ-excessive functions -- Brownian local time -- 17. Proof of the Volkonskii-Šur-Meyer Theorem -- 18. Killing -- 19. The Feynmann-Kac formula -- 20. A Ciesielski-Taylor Theorem -- 21. Time-substitution -- 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 -- 5. Ray Processes -- 33. Orientation -- 34. Ray resolvents -- 35. The Ray-Knight compactification -- Ray's Theorem: analytical part -- 36. From semigroup to resolvent -- 37. Branch-points -- 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 -- 6. Applications -- 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 -- 45. Doob h-transforms -- Time reversal and related topics -- 46. Nagasawa's formula for chains -- 47. Strong Markov property under time reversal -- 48. Equilibrium charge -- 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)]. , English

Additional Edition: ISBN 0-521-77594-9

Language: English

URL: https://doi.org/10.1017/CBO9781107590120