Diffusions, Markov processes, and martingales / David Williams.
Series Wiley series in probability and mathematical statisticsEditor: Chichester : Wiley, c1979-c1987Descripción: 2 v. ; 24 cmISBN: 0471997056 (v. 1); 0471914827 (v. 2)Tema(s): Markov processes | Diffusion processes | Martingales (Mathematics)Otra clasificación: 60J25 | 60J60 (60G07 60H05 60J25)Vol. 1 CHAPTER I. INTRODUCTION TO BROWNIAN MOTION -- 1. WIENER MEASURE (§§ 1-29) [1] -- Wiener’s theorem (§ 6); A converse (9); Diffusions (10); Canonical and non-canonical processes (11-14); Martingale descriptions of Brownian motion (15-17); Levy’s theorem (17); Trotter’s theorem (18-20); Local and global properties (21-26); Blumenthal’s 01 law (23); Kolmogorov’s test (24); Iterated-logarithm law (25); Holder condition (26); Completions, almost surely (P) (27-29). -- 2. NARROW CONVERGENCE (30-52) [18] -- Donsker invariance principle (30-39); Polish spaces (36); Narrow convergence of measures (37); Prohorov’s theorem (40-47); Narrow convergence in Pr( W) (48-52). -- 3. BROWNIAN MOTION IN R" (53-66) [29] -- Potential theory (58-62); Equilibrium potential (59); Bessel processes (63-66); Skew product (64); Ray-Knight theorem on local time (65). -- CHAPTER II. SOME CLASSICAL THEORY -- 1. BASIC MEASURE THEORY (1-14) [39] -- Monotone-class theorems (1-5, 13—14); Daniell-Kolmogorov theorem (6); Its limitations (7); Fubini’s theorem (8); Infinite products (9); Poisson measures (10); Stochastic process (11); Modification (12). -- 2. CLASSICAL MARTINGALE THEORY (15-55) [50] -- Uniform integrability (15-20); Conditional expectations and probabilities (21-24); Discrete-parameter martingales (23-34); Continuous-parameter supermartingales (35-55); Basic definitions (36); Skorokhod (càdlàg) maps (37); Doob’s regularity theorem (38-41); The ‘usual conditions’ (40); Indistinguishability, evanescence (42); Inequalities and the convergence theorem (43); Stopping times (44-50); Debut and section theorems (51-52); Stopping times and supcrmartingales (53-55). -- 3. APPLICATIONS (56-67) [81] -- Proof of Wiener's theorem (56); Strong Markov theorem for Brownian motion (57); Hitting-times, reflection principle, etc. (58— 61); Levy's downcrossing theorem (62); Local time (63); Hitting-time process as subordinator (64); Pitman’s presentation of 3-dimensional Bessel process (65); Excursion theory (66-67). -- 4 REGULAR CONDITIONAL PROBABILITIES (68-72) [100] -- Mam theorem (69); Fundamental statements of Markov property (72). -- CHAPTER III. MARKOV PROCESSES -- 1. TRANSITION FUNCTIONS AND RESOLVENTS (1-6) [106] -- Definitions (1—2); Hille-Yosida theorems (3-6) -- 2. FELLER TRANSITION FUNCTIONS (7-10) [114] -- Feller—Dynkin (FD) semigroups (8); Dynkin’s maximum principle -- 3. FELLER-DYNKIN PROCESSES (11-31) [117] -- Path regularization (13); Canonical FD process (14); Strong Markov theorem for FD processes (15—17); Completions (16); Blumenthal’s 01 law (18); Some fundamental martingales, Dynkin’s formula (22); Quasi-left-continuity (23); Characteristic operator (24); FD diffusions (26—29); Dirichlet problem (30). -- 4. ADDITIVE FUNCTIONALS (32-42) [141] -- Some basic facts about PCHAFs (32—34); Killing (35-36); Timesubstitution (37); Volkonskii’s formula. Arcsine law, Feller-McKean chain (38); Feynman—Kac formula (39); A Ciesielski-Taylor theorem (40); Elastic Brownian motion (41). -- 5. RAY PROCESSES (43-66) [162] -- Motivation (43—54); Martin boundary theory for discrete-parameter chains (48); Probabilistic Doob-Hunt theory (discrete-parameter chains) (49); R. S. Martin’s boundary (50); Doob-Hunt theory for Brownian motion (51); Ray’s theorem: preparatory remarks (55-56): Ray—Knight compactification (57); Ray resolvents (58); Ray’s theorem: analytic part (59-62); Branch-points (60); Ray’s theorem: probabilistic part (63-66); Strong Markov theorem for Ray processes (64); The r61e of branch-points (65). -- 6. APPLICATIONS (67-91) [198] -- Martin boundary theory in retrospect (68-73); Proof of the Doob-Hunt convergence theorem (70); Choquet representation of excessive functions (71); Doob’s A-transforms (72); Time-reversal and related topics (74-79); Nagasawa’s formula (75); Strong Markov property under time-reversal (76); Equilibrium charge (77); Split-ting-times (79); A first look at chain theory (80-91); Chains as Ray processes (81); Taboo probabilities; first-entrance decompositions (83); The Q-matrix: DK conditions (84); Local character condition for Q (85); Totally instantaneous Q-matrices (86); Last exits (87); Excursions (88); Kingman’s solution of the Markov characterization problem (89); Q-matrix problem: symmetrizable case (90). -- REFERENCES [229] -- INDEX [235] --
Vol. 2 CHAPTER IV. INTRODUCTION TO ITO CALCULUS -- TERMINOLOGY AND CONVENTIONS -- R-processes and L-processes -- Usual conditions -- Important convention about time [0] -- 1. SOME MOTIVATING REMARKS -- 1. Ito integrals................... [2] -- 2 Integration by parts ................ [4] -- 3. Ito’s formula for Brownian motion ...........8 -- 4. A rough plan of the chapter ............. [9] -- 2 SOME FUNDAMENTAL IDEAS: PREVISIBLE PROCESSES, LOCALIZATION, etc. -- Previsible processes -- 5. Basic integrands Z(S, 7*J ..............10 -- 6. Previsible processes on ................11 -- Finite-variation and integrable-variation processes -- 7. FV0 and IV0 processes ...............14 -- 8. Preservation of the martingale property .........14 -- Localization -- 9. H(0, T],Xr...................................15 -- 10. Localization of integrands, Ib...............16 -- 11. Localization of integrators, FV^T etc........17 -- 12. Nil desperandum!.............................18 -- 13. Extending stochastic integrals by localization [20] -- 14. Local martingales, and the Fatou lemma.........21 -- Semimartingales as integrators -- 15. Semimartingales,...............................23 -- 16. Integrators....................................24 -- Likelihood ratios -- 17. Martingale property under change of measure.....25 -- 3. THE ELEMENTARY THEORY OF FINITE-VARIATION PROCESSES -- 18. Ito’s formula for FV functions [27] -- 19. The Doleans exponential F (x.) ............ [29] -- Applications to Markov chains with finite state-space -- 20. Martingale problems................ [30] -- 21. Probabilistic interpretation of Q....... [33] -- 22. Likelihood ratios and some key distributions ....... [37] -- 4. STOCHASTIC INTEGRALS: THE L2 THEORY -- 23. Orientation ................... [42] -- 24. Stable spaces of ........... [42] -- 25. Elementary stochastic integrals relative to . . . . [45] -- 26. The processes ............ [46] -- 27. Constructing stochastic integrals in L2 [47] -- 28. The Kunita-Watanabe inequalities........... [50] -- 5. STOCHASTIC INTEGRALS WITH RESPECT TO -- CONTINUOUS SEMIMARTINGALES -- 29. Orientation ................... [52] -- 30. Quadratic variation for continuous local martingales . . . . [52] -- 31. Canonical decomposition of a continuous semimartingale. . . [57] -- 32. Ito’s formula for continuous semimartingales ....... [58] -- 6. APPLICATIONS OF ITO’s FORMULA -- 33. Levy’s theorem........... . ...... [63] -- 34. Continuous local martingales as time-changes of Brownian -- motion . [64] -- 35. Bessel processes; skew products; etc. [69] -- 36. Brownian martingale representation ......... [73] -- 37. Exponential semimartingales; estimates ....... . . [75] -- 38. Cameron-Martin-Girsanov change of measure ...... [79] -- 39. First applications: Doob ^-transforms; hitting of spheres; etc. . [83] -- 40. Further applications: bridges; excursions; etc........ [86] -- 41. Explicit Brownian martingale representation ....... [89] -- 42. Burkholder-Davis-Gundy inequalities . . . ...... [93] -- 43. Semimartingale local time; Tanaka's formula ....... [95] -- 44. Study of joint continuity ........... . . . [99] -- 45. Local time as an occupation density; generalized Ito-Tanaka -- formula . . [102] -- 46. The Stratonovich calculus [106] -- 47. Ricmann-sum approximation to ltd and Stratonovich integrals; -- simulation [108] -- CHAPTER V. STOCHASTIC DIFFERENTIAL EQUATIONS AND DIFFUSIONS -- I. INTRODUCTION -- I. What is a diffusion in R*? [110] -- 2 FD diffusions recalled [112] -- 3. SDEs as a means of constructing diffusions ........ [113] -- 4 Example: Brownian motion on a surface [114] -- 5. Examples: modelling noise in physical systems....... [114] -- 6. Example: Skorokhod’s equation ............ I17 -- 7. Examples: control problems ............. [119] -- 2 PATHWISE UNIQUENESS, STRONG SDEs, FLOWS -- 8. Our general SDE; previsible path functionals; diffusion SDEs [122] -- 9. Pathwise uniqueness; exact SDEs ........... [124] -- 10. Relationship between exact SDEs and strong solutions. . . . [125] -- 11. The Ito existence and uniqueness result ......... [128] -- 12 Locally Lipschitz SDEs; Lipschitz properties of a,1/2 . [132] -- 13. Flows; the diffeomorphism theorem; time-reversed flows . . . [136] -- 14. Carverhill’s noisy North-South flow on a circle ...... [141] -- 15. The martingale optimality principle in control....... [144] -- 3. WEAK SOLUTIONS, UNIQUENESS IN LAW -- 16. Weak solutions of SDEs; Tanaka's SDE ......... [149] -- 17. ‘Exact equals weak plus pathwise unique* ........ [151] -- 18. Tsirel’son’s example ................ [155] -- 4. MARTINGALE PROBLEMS, MARKOV PROPERTY -- 19. Definition; orientation ............... [158] -- 20. Equivalence of the martingale-problem and ‘weak’ formulations [160] -- 21. Martingale problems and the strong Markov property . . . . 162 22 Appraisal and consolidation: where we have reached . . . . [163] -- 23. Existence of solutions to the martingale problem ...... [166] -- 24. The Stroock-Varadhan uniqueness theorem ....... [170] -- 25. Martingale representation ........... . [173] -- Transformation of SDEs -- 26. Change of time scale; Girsanov’s SDE [175] -- 27. Change of measure ......... [177] -- 28. Change of state-space; scale; Zvonkin’s observation; the Doss- -- Sussmann method................. [178] -- 29. Krylov's example . [181] -- 5. OVERTURE TO STOCHASTIC DIFFERENTIAL GEOMETRY -- 30. Introduction; some key ideas; Stratonovich-to-Itd conversion [182] -- 31. Brownian motion on a submanifold of R" [186] -- CHAPTER V. STOCHASTIC DIFFERENTIAL EQUATIONS AND DIFFUSIONS -- I. INTRODUCTION -- I. What is a diffusion in R*? [110] -- 2 FD diffusions recalled [112] -- 3. SDEs as a means of constructing diffusions ........ [113] -- 4 Example: Brownian motion on a surface [114] -- 5. Examples: modelling noise in physical systems....... [114] -- 6. Example: Skorokhod’s equation ............ I17 -- 7. Examples: control problems ............. [119] -- 2 PATHWISE UNIQUENESS, STRONG SDEs, FLOWS -- 8. Our general SDE; previsible path functionals; diffusion SDEs [122] -- 9. Pathwise uniqueness; exact SDEs ........... [124] -- 10. Relationship between exact SDEs and strong solutions. . . . [125] -- 11. The Ito existence and uniqueness result ......... [128] -- 12 8. ITO EXCURSION THEORY -- Locally Lipschitz SDEs; Lipschitz properties of a,1/2 . [132] -- 13. Flows; the diffeomorphism theorem; time-reversed flows . . . [136] -- 14. Carverhill’s noisy North-South flow on a circle ...... [141] -- 15. The martingale optimality principle in control....... [144] -- 3. WEAK SOLUTIONS, UNIQUENESS IN LAW -- 16. Weak solutions of SDEs; Tanaka's SDE ......... [149] -- 17. ‘Exact equals weak plus pathwise unique* ........ [151] -- 18. Tsirel’son’s example ................ [155] -- 4. MARTINGALE PROBLEMS, MARKOV PROPERTY -- 19. Definition; orientation ............... [158] -- 20. Equivalence of the martingale-problem and ‘weak’ formulations [160] -- 21. Martingale problems and the strong Markov property . . . . 162 22 Appraisal and consolidation: where we have reached . . . . [163] -- 23. Existence of solutions to the martingale problem ...... [166] -- 24. The Stroock-Varadhan uniqueness theorem ....... [170] -- 25. Martingale representation ........... . [173] -- Transformation of SDEs -- 26. Change of time scale; Girsanov’s SDE [175] -- 27. Change of measure ......... [177] -- 28. Change of state-space; scale; Zvonkin’s observation; the Doss- -- Sussmann method................. [178] -- 29. Krylov's example . [181] -- 5. OVERTURE TO STOCHASTIC DIFFERENTIAL GEOMETRY -- 30. Introduction; some key ideas; Stratonovich-to-Itd conversion [182] -- 31. Brownian motion on a submanifold of R" [186] -- 32. Parallel displacement; Riemannian connections [193] -- 33. Extrinsic theory of BM^'fOfl)); rolling without slipping; martingales on manifolds; etc. [198] -- 34. Intrinsic theory; normal coordinates; structural equations; diffusions on manifolds; etc. [203] -- 35. Brownian motion on Lie groups ...... [224] -- 36. Dynkin’s Brownian motion of ellipses; hyperbolic space interpretation; etc. . . [239] -- 37. Khasminskii's method for studying stability; random vibrations. [246] -- 38. Hormander’s theorem; Malliavin calculus; stochastic pullback; -- curvature.................. [250] -- 6. ONE-DIMENSIONAL SDEs -- 39. A local-time criterion for pathwise uniqueness [263] -- 40. The Yamada-Watanabe pathwise uniqueness theorem .... [265] -- 41. The Nakao pathwise-uniqueness theorem ........ [266] -- 42. Solution of a variance control problem ......... [267] -- 43. A comparison theorem ............... [269] -- 7. ONE-DIMENSIONAL DIFFUSIONS -- 44. Orientation ................... [270] -- 45. Regular diffusions. [271] -- 46. The scale function, s ................ [273] -- 47. The speed measure, m; time substitution ......... [276] -- 48. Example: the Bessel SDE . * ............. [284] -- 49. Diffusion local time ........... '..... [289] -- 50. Analytical aspects................. [291] -- 51. Classification of boundary points ........... [295] -- 52. Khasminskii’s test for explosions ........... [297] -- 53. An ergodic theorem for 1-dimensional diffusions ...... [300] -- 54. Coupling of 1-dimensional diffusions .......... [301] -- CHAPTER VI. THE GENERAL THEORY -- 1 ORIENTATION -- 1. Preparatory remarks [304] -- 2. Levy processes [308] -- 2. DEBUT AND SECTION THEOREMS -- 3. Progressive processes ............... [313] -- 4. Optional processes,^; optional times. . . . . . . . . . [315] -- 5. The ‘optional’ section theorem ............ [317] -- 6. Wanting (not to be skipped) ....... ...... [318] -- 3. OPTIONAL PROJECTIONS AND FILTERING -- 7. Optional projection "X of X -- 8. The innovations approach to filtering -- 9. The Kalman- Bucy filter -- 10. The Bayesian approach to filtering; a change-detection filter . . -- 11. Robust filtering -- 4. CHARACTERIZING PREVISIBLE TIMES -- 12. Previsible stopping times; PFA theorem ......... -- 13. Totally inaccessible and accessible stopping times -- 14. Some examples.................. -- 15. Meyer’s previsibility theorem for Markov processes . . . . . -- 16. Proof of the PFA theorem.............. -- 17. The a-algebras -- 18. Quasi-left-continuous filtrations ............ -- 5. DUAL PREVISIBLE PROJECTIONS -- 19. The previsible section theorem; the previsible projection pX -- of x ..................... [347] -- 20. Doleans’ characterization of FV processes [349] -- 21. Dual previsible projections, compensators ........ [350] -- 22. Cumulative risk ................. [352] -- 23. Some Brownian motion examples ........... [354] -- 24. Decomposition of a continuous semimartingale ...... [358] -- 25. Proof of the basic (u, A) correspondence ......... [359] -- 26. Proof of the Doleans ‘optional’ characterization result . . . . [360] -- 27. Proof of the Doleans ‘previsible’ characterization result . . [361] -- 28. Levy systems for Markov processes [364] -- 6. THE MEYER DECOMPOSITION THEOREM -- 29. Introduction .' ....... [367] -- 30. The Doleans proof of the Meyer decomposition ...... [369] -- 31. Regular class (D) submartingales; approximation to compensators [372] -- 32. The local form of the decomposition theorem [374] -- 33. An L2 bounded local martingale which is not a martingale . . [375] -- 34. The process ................. [376] -- 35. Last exits and equilibrium charge ........... [377] -- 7. STOCHASTIC INTEGRATION: THE GENERAL CASE -- 36. The quadratic variation process [M] .......... [382] -- 37. Stochastic integrals with respect to local martingales .... [388] -- 38. Stochastic integrals with respect to semimartingales [391] -- 39. Ito’s formula for semimartingales [394] -- 40. Special semimartingales [394] -- 41. Quasimartingales [396] -- 8. ITO EXCURSION TEHORY -- 42. Introduction ............. [398] -- 43. Exursion theory for a finite Markov chain [400] -- 44. Taking stock [405] -- 45. Local time L at a regular extremal point a [406] -- 46. Some technical point* hypotheses drones, etc ...... [410] -- 47. The Portion point proem of excursions [413] -- 48. Markovian character of a [414] -- 49. Marking the excursions............... [418] -- 50. Last-exit decomposition, calculation of the excursion law n . . [420] -- 51. The Skorokhod embedding theorem .......... [423] -- 52. Diffusion properties of local time in the space vanable; the -- Ray Knight theorem [428] -- 53. Arcsine law for Brownian motion ........... [431] -- 54. Resolvent density of a 1-dimensional diffusion ....... [432] -- 55. Path decomposition of Brownian motions and of excursions. . [433] -- 56. An illustrative calculation .............. [438] -- 57. Feller Brownian motions .............. [439] -- 58. Example: censoring and reweighting of excursion laws . . . . [442] -- 59. Excursion theory by stochastic calculus: McGill’s lemma . . . [445] -- REFERENCES.................... [449] -- INDEX....................... [469] --
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 60 W721 (Browse shelf) | Vol. 2 | Available | A-6434 |
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Autores del vol. 2: L. C. G. Rogers and David Williams.
Incluye referencias bibliográficas e índices.
Vol. 1
CHAPTER I. INTRODUCTION TO BROWNIAN MOTION --
1. WIENER MEASURE (§§ 1-29) [1] --
Wiener’s theorem (§ 6); A converse (9); Diffusions (10); Canonical and non-canonical processes (11-14); Martingale descriptions of Brownian motion (15-17); Levy’s theorem (17); Trotter’s theorem (18-20); Local and global properties (21-26); Blumenthal’s 01 law (23); Kolmogorov’s test (24); Iterated-logarithm law (25); Holder condition (26); Completions, almost surely (P) (27-29). --
2. NARROW CONVERGENCE (30-52) [18] --
Donsker invariance principle (30-39); Polish spaces (36); Narrow convergence of measures (37); Prohorov’s theorem (40-47); Narrow convergence in Pr( W) (48-52). --
3. BROWNIAN MOTION IN R" (53-66) [29] --
Potential theory (58-62); Equilibrium potential (59); Bessel processes (63-66); Skew product (64); Ray-Knight theorem on local time (65). --
CHAPTER II. SOME CLASSICAL THEORY --
1. BASIC MEASURE THEORY (1-14) [39] --
Monotone-class theorems (1-5, 13—14); Daniell-Kolmogorov theorem (6); Its limitations (7); Fubini’s theorem (8); Infinite products (9); Poisson measures (10); Stochastic process (11); Modification (12). --
2. CLASSICAL MARTINGALE THEORY (15-55) [50] --
Uniform integrability (15-20); Conditional expectations and probabilities (21-24); Discrete-parameter martingales (23-34); Continuous-parameter supermartingales (35-55); Basic definitions (36); Skorokhod (càdlàg) maps (37); Doob’s regularity theorem (38-41); The ‘usual conditions’ (40); Indistinguishability, evanescence (42); Inequalities and the convergence theorem (43); Stopping times (44-50); Debut and section theorems (51-52); Stopping times and supcrmartingales (53-55). --
3. APPLICATIONS (56-67) [81] --
Proof of Wiener's theorem (56); Strong Markov theorem for Brownian motion (57); Hitting-times, reflection principle, etc. (58— 61); Levy's downcrossing theorem (62); Local time (63); Hitting-time process as subordinator (64); Pitman’s presentation of 3-dimensional Bessel process (65); Excursion theory (66-67). --
4 REGULAR CONDITIONAL PROBABILITIES (68-72) [100] --
Mam theorem (69); Fundamental statements of Markov property (72). --
CHAPTER III. MARKOV PROCESSES --
1. TRANSITION FUNCTIONS AND RESOLVENTS (1-6) [106] --
Definitions (1—2); Hille-Yosida theorems (3-6) --
2. FELLER TRANSITION FUNCTIONS (7-10) [114] --
Feller—Dynkin (FD) semigroups (8); Dynkin’s maximum principle --
3. FELLER-DYNKIN PROCESSES (11-31) [117] --
Path regularization (13); Canonical FD process (14); Strong Markov theorem for FD processes (15—17); Completions (16); Blumenthal’s 01 law (18); Some fundamental martingales, Dynkin’s formula (22); Quasi-left-continuity (23); Characteristic operator (24); FD diffusions (26—29); Dirichlet problem (30). --
4. ADDITIVE FUNCTIONALS (32-42) [141] --
Some basic facts about PCHAFs (32—34); Killing (35-36); Timesubstitution (37); Volkonskii’s formula. Arcsine law, Feller-McKean chain (38); Feynman—Kac formula (39); A Ciesielski-Taylor theorem (40); Elastic Brownian motion (41). --
5. RAY PROCESSES (43-66) [162] --
Motivation (43—54); Martin boundary theory for discrete-parameter chains (48); Probabilistic Doob-Hunt theory (discrete-parameter chains) (49); R. S. Martin’s boundary (50); Doob-Hunt theory for Brownian motion (51); Ray’s theorem: preparatory remarks (55-56): Ray—Knight compactification (57); Ray resolvents (58); Ray’s theorem: analytic part (59-62); Branch-points (60); Ray’s theorem: probabilistic part (63-66); Strong Markov theorem for Ray processes (64); The r61e of branch-points (65). --
6. APPLICATIONS (67-91) [198] --
Martin boundary theory in retrospect (68-73); Proof of the Doob-Hunt convergence theorem (70); Choquet representation of excessive functions (71); Doob’s A-transforms (72); Time-reversal and related topics (74-79); Nagasawa’s formula (75); Strong Markov property under time-reversal (76); Equilibrium charge (77); Split-ting-times (79); A first look at chain theory (80-91); Chains as Ray processes (81); Taboo probabilities; first-entrance decompositions (83); The Q-matrix: DK conditions (84); Local character condition for Q (85); Totally instantaneous Q-matrices (86); Last exits (87); Excursions (88); Kingman’s solution of the Markov characterization problem (89); Q-matrix problem: symmetrizable case (90). --
REFERENCES [229] --
INDEX [235] --
Vol. 2
CHAPTER IV. INTRODUCTION TO ITO CALCULUS --
TERMINOLOGY AND CONVENTIONS --
R-processes and L-processes --
Usual conditions --
Important convention about time [0] --
1. SOME MOTIVATING REMARKS --
1. Ito integrals................... [2] --
2 Integration by parts ................ [4] --
3. Ito’s formula for Brownian motion ...........8 --
4. A rough plan of the chapter ............. [9] --
2 SOME FUNDAMENTAL IDEAS: PREVISIBLE PROCESSES, LOCALIZATION, etc. --
Previsible processes --
5. Basic integrands Z(S, 7*J ..............10 --
6. Previsible processes on ................11 --
Finite-variation and integrable-variation processes --
7. FV0 and IV0 processes ...............14 --
8. Preservation of the martingale property .........14 --
Localization --
9. H(0, T],Xr...................................15 --
10. Localization of integrands, Ib...............16 --
11. Localization of integrators, FV^T etc........17 --
12. Nil desperandum!.............................18 --
13. Extending stochastic integrals by localization [20] --
14. Local martingales, and the Fatou lemma.........21 --
Semimartingales as integrators --
15. Semimartingales,...............................23 --
16. Integrators....................................24 --
Likelihood ratios --
17. Martingale property under change of measure.....25 --
3. THE ELEMENTARY THEORY OF FINITE-VARIATION PROCESSES --
18. Ito’s formula for FV functions [27] --
19. The Doleans exponential F (x.) ............ [29] --
Applications to Markov chains with finite state-space --
20. Martingale problems................ [30] --
21. Probabilistic interpretation of Q....... [33] --
22. Likelihood ratios and some key distributions ....... [37] --
4. STOCHASTIC INTEGRALS: THE L2 THEORY --
23. Orientation ................... [42] --
24. Stable spaces of ........... [42] --
25. Elementary stochastic integrals relative to . . . . [45] --
26. The processes ............ [46] --
27. Constructing stochastic integrals in L2 [47] --
28. The Kunita-Watanabe inequalities........... [50] --
5. STOCHASTIC INTEGRALS WITH RESPECT TO --
CONTINUOUS SEMIMARTINGALES --
29. Orientation ................... [52] --
30. Quadratic variation for continuous local martingales . . . . [52] --
31. Canonical decomposition of a continuous semimartingale. . . [57] --
32. Ito’s formula for continuous semimartingales ....... [58] --
6. APPLICATIONS OF ITO’s FORMULA --
33. Levy’s theorem........... . ...... [63] --
34. Continuous local martingales as time-changes of Brownian --
motion . [64] --
35. Bessel processes; skew products; etc. [69] --
36. Brownian martingale representation ......... [73] --
37. Exponential semimartingales; estimates ....... . . [75] --
38. Cameron-Martin-Girsanov change of measure ...... [79] --
39. First applications: Doob ^-transforms; hitting of spheres; etc. . [83] --
40. Further applications: bridges; excursions; etc........ [86] --
41. Explicit Brownian martingale representation ....... [89] --
42. Burkholder-Davis-Gundy inequalities . . . ...... [93] --
43. Semimartingale local time; Tanaka's formula ....... [95] --
44. Study of joint continuity ........... . . . [99] --
45. Local time as an occupation density; generalized Ito-Tanaka --
formula . . [102] --
46. The Stratonovich calculus [106] --
47. Ricmann-sum approximation to ltd and Stratonovich integrals; --
simulation [108] --
CHAPTER V. STOCHASTIC DIFFERENTIAL EQUATIONS AND DIFFUSIONS --
I. INTRODUCTION --
I. What is a diffusion in R*? [110] --
2 FD diffusions recalled [112] --
3. SDEs as a means of constructing diffusions ........ [113] --
4 Example: Brownian motion on a surface [114] --
5. Examples: modelling noise in physical systems....... [114] --
6. Example: Skorokhod’s equation ............ I17 --
7. Examples: control problems ............. [119] --
2 PATHWISE UNIQUENESS, STRONG SDEs, FLOWS --
8. Our general SDE; previsible path functionals; diffusion SDEs [122] --
9. Pathwise uniqueness; exact SDEs ........... [124] --
10. Relationship between exact SDEs and strong solutions. . . . [125] --
11. The Ito existence and uniqueness result ......... [128] --
12 Locally Lipschitz SDEs; Lipschitz properties of a,1/2 . [132] --
13. Flows; the diffeomorphism theorem; time-reversed flows . . . [136] --
14. Carverhill’s noisy North-South flow on a circle ...... [141] --
15. The martingale optimality principle in control....... [144] --
3. WEAK SOLUTIONS, UNIQUENESS IN LAW --
16. Weak solutions of SDEs; Tanaka's SDE ......... [149] --
17. ‘Exact equals weak plus pathwise unique* ........ [151] --
18. Tsirel’son’s example ................ [155] --
4. MARTINGALE PROBLEMS, MARKOV PROPERTY --
19. Definition; orientation ............... [158] --
20. Equivalence of the martingale-problem and ‘weak’ formulations [160] --
21. Martingale problems and the strong Markov property . . . . 162 22 Appraisal and consolidation: where we have reached . . . . [163] --
23. Existence of solutions to the martingale problem ...... [166] --
24. The Stroock-Varadhan uniqueness theorem ....... [170] --
25. Martingale representation ........... . [173] --
Transformation of SDEs --
26. Change of time scale; Girsanov’s SDE [175] --
27. Change of measure ......... [177] --
28. Change of state-space; scale; Zvonkin’s observation; the Doss- --
Sussmann method................. [178] --
29. Krylov's example . [181] --
5. OVERTURE TO STOCHASTIC DIFFERENTIAL GEOMETRY --
30. Introduction; some key ideas; Stratonovich-to-Itd conversion [182] --
31. Brownian motion on a submanifold of R" [186] --
CHAPTER V. STOCHASTIC DIFFERENTIAL EQUATIONS AND DIFFUSIONS --
I. INTRODUCTION --
I. What is a diffusion in R*? [110] --
2 FD diffusions recalled [112] --
3. SDEs as a means of constructing diffusions ........ [113] --
4 Example: Brownian motion on a surface [114] --
5. Examples: modelling noise in physical systems....... [114] --
6. Example: Skorokhod’s equation ............ I17 --
7. Examples: control problems ............. [119] --
2 PATHWISE UNIQUENESS, STRONG SDEs, FLOWS --
8. Our general SDE; previsible path functionals; diffusion SDEs [122] --
9. Pathwise uniqueness; exact SDEs ........... [124] --
10. Relationship between exact SDEs and strong solutions. . . . [125] --
11. The Ito existence and uniqueness result ......... [128] --
12 8. ITO EXCURSION THEORY --
Locally Lipschitz SDEs; Lipschitz properties of a,1/2 . [132] --
13. Flows; the diffeomorphism theorem; time-reversed flows . . . [136] --
14. Carverhill’s noisy North-South flow on a circle ...... [141] --
15. The martingale optimality principle in control....... [144] --
3. WEAK SOLUTIONS, UNIQUENESS IN LAW --
16. Weak solutions of SDEs; Tanaka's SDE ......... [149] --
17. ‘Exact equals weak plus pathwise unique* ........ [151] --
18. Tsirel’son’s example ................ [155] --
4. MARTINGALE PROBLEMS, MARKOV PROPERTY --
19. Definition; orientation ............... [158] --
20. Equivalence of the martingale-problem and ‘weak’ formulations [160] --
21. Martingale problems and the strong Markov property . . . . 162 22 Appraisal and consolidation: where we have reached . . . . [163] --
23. Existence of solutions to the martingale problem ...... [166] --
24. The Stroock-Varadhan uniqueness theorem ....... [170] --
25. Martingale representation ........... . [173] --
Transformation of SDEs --
26. Change of time scale; Girsanov’s SDE [175] --
27. Change of measure ......... [177] --
28. Change of state-space; scale; Zvonkin’s observation; the Doss- --
Sussmann method................. [178] --
29. Krylov's example . [181] --
5. OVERTURE TO STOCHASTIC DIFFERENTIAL GEOMETRY --
30. Introduction; some key ideas; Stratonovich-to-Itd conversion [182] --
31. Brownian motion on a submanifold of R" [186] --
32. Parallel displacement; Riemannian connections [193] --
33. Extrinsic theory of BM^'fOfl)); rolling without slipping; martingales on manifolds; etc. [198] --
34. Intrinsic theory; normal coordinates; structural equations; diffusions on manifolds; etc. [203] --
35. Brownian motion on Lie groups ...... [224] --
36. Dynkin’s Brownian motion of ellipses; hyperbolic space interpretation; etc. . . [239] --
37. Khasminskii's method for studying stability; random vibrations. [246] --
38. Hormander’s theorem; Malliavin calculus; stochastic pullback; --
curvature.................. [250] --
6. ONE-DIMENSIONAL SDEs --
39. A local-time criterion for pathwise uniqueness [263] --
40. The Yamada-Watanabe pathwise uniqueness theorem .... [265] --
41. The Nakao pathwise-uniqueness theorem ........ [266] --
42. Solution of a variance control problem ......... [267] --
43. A comparison theorem ............... [269] --
7. ONE-DIMENSIONAL DIFFUSIONS --
44. Orientation ................... [270] --
45. Regular diffusions. [271] --
46. The scale function, s ................ [273] --
47. The speed measure, m; time substitution ......... [276] --
48. Example: the Bessel SDE . * ............. [284] --
49. Diffusion local time ........... '..... [289] --
50. Analytical aspects................. [291] --
51. Classification of boundary points ........... [295] --
52. Khasminskii’s test for explosions ........... [297] --
53. An ergodic theorem for 1-dimensional diffusions ...... [300] --
54. Coupling of 1-dimensional diffusions .......... [301] --
CHAPTER VI. THE GENERAL THEORY --
1 ORIENTATION --
1. Preparatory remarks [304] --
2. Levy processes [308] --
2. DEBUT AND SECTION THEOREMS --
3. Progressive processes ............... [313] --
4. Optional processes,^; optional times. . . . . . . . . . [315] --
5. The ‘optional’ section theorem ............ [317] --
6. Wanting (not to be skipped) ....... ...... [318] --
3. OPTIONAL PROJECTIONS AND FILTERING --
7. Optional projection "X of X --
8. The innovations approach to filtering --
9. The Kalman- Bucy filter --
10. The Bayesian approach to filtering; a change-detection filter . . --
11. Robust filtering --
4. CHARACTERIZING PREVISIBLE TIMES --
12. Previsible stopping times; PFA theorem ......... --
13. Totally inaccessible and accessible stopping times --
14. Some examples.................. --
15. Meyer’s previsibility theorem for Markov processes . . . . . --
16. Proof of the PFA theorem.............. --
17. The a-algebras --
18. Quasi-left-continuous filtrations ............ --
5. DUAL PREVISIBLE PROJECTIONS --
19. The previsible section theorem; the previsible projection pX --
of x ..................... [347] --
20. Doleans’ characterization of FV processes [349] --
21. Dual previsible projections, compensators ........ [350] --
22. Cumulative risk ................. [352] --
23. Some Brownian motion examples ........... [354] --
24. Decomposition of a continuous semimartingale ...... [358] --
25. Proof of the basic (u, A) correspondence ......... [359] --
26. Proof of the Doleans ‘optional’ characterization result . . . . [360] --
27. Proof of the Doleans ‘previsible’ characterization result . . [361] --
28. Levy systems for Markov processes [364] --
6. THE MEYER DECOMPOSITION THEOREM --
29. Introduction .' ....... [367] --
30. The Doleans proof of the Meyer decomposition ...... [369] --
31. Regular class (D) submartingales; approximation to compensators [372] --
32. The local form of the decomposition theorem [374] --
33. An L2 bounded local martingale which is not a martingale . . [375] --
34. The process ................. [376] --
35. Last exits and equilibrium charge ........... [377] --
7. STOCHASTIC INTEGRATION: THE GENERAL CASE --
36. The quadratic variation process [M] .......... [382] --
37. Stochastic integrals with respect to local martingales .... [388] --
38. Stochastic integrals with respect to semimartingales [391] --
39. Ito’s formula for semimartingales [394] --
40. Special semimartingales [394] --
41. Quasimartingales [396] --
8. ITO EXCURSION TEHORY --
42. Introduction ............. [398] --
43. Exursion theory for a finite Markov chain [400] --
44. Taking stock [405] --
45. Local time L at a regular extremal point a [406] --
46. Some technical point* hypotheses drones, etc ...... [410] --
47. The Portion point proem of excursions [413] --
48. Markovian character of a [414] --
49. Marking the excursions............... [418] --
50. Last-exit decomposition, calculation of the excursion law n . . [420] --
51. The Skorokhod embedding theorem .......... [423] --
52. Diffusion properties of local time in the space vanable; the --
Ray Knight theorem [428] --
53. Arcsine law for Brownian motion ........... [431] --
54. Resolvent density of a 1-dimensional diffusion ....... [432] --
55. Path decomposition of Brownian motions and of excursions. . [433] --
56. An illustrative calculation .............. [438] --
57. Feller Brownian motions .............. [439] --
58. Example: censoring and reweighting of excursion laws . . . . [442] --
59. Excursion theory by stochastic calculus: McGill’s lemma . . . [445] --
REFERENCES.................... [449] --
INDEX....................... [469] --
MR, 80i:60100 (v. 1)
MR, 89k:60117 (v. 2)
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