An essay on the general theory of stochastic processes

Probability Surveys

Volumn 3, Issue 1, 2006, Pages 345-412

  Nikeghbali, Ashkan ^a  

^a ETH ZURICH   (Switzerland)

Author keywords

Enlarge ments of filtrations; General theory of stochastic processes; Hon est times; Pseudo stopping times; Random times; Stopping times; Submartingales

Indexed keywords

EID: 33847407852     PISSN: None     EISSN: 15495787     Source Type: Journal    
DOI: 10.1214/154957806000000104     Document Type: Article

References (81)

1. G.E. Andrews, R. Askey, R. Roy: Special functions, Cambridge University Press (1999). MR1688958
2. J. Azéma: Quelques applications de la théorie générale des processus I, Invent. Math. 18 (1972) 293-336. MR0326848
3. J. Azéma, Représentation d'une surmartingale bornée, ZW, 45, 1978, 191-212. MR0510025
4. J. Azéma: Sur les fermés aléatoires, Sém.Proba. XIX, Lecture Notes in Mathematics 1123, (1985), 397-495. MR0889496
5. J. Azéma, T. Jeulin, F. Knight, G. Mokobodzki, M. Yor: Sur les processus croissants de type injectif, Sém.Proba. XXX, Lecture Notes in Mathematics 1626, (1996), 312-343. MR1459491
6. J. Azéma, T. Jeulin, F. Knight, M. Yor: Le théorème d'arr̂ et en une fin d'ensemble prévisible, Sém.Proba. XXVII, Lecture Notes in Mathematics 1557, (1993), 133-158. MR1308560
7. J. Azéma, T. Jeulin, F. Knight, M. Yor: Quelques calculs de compen-sateurs impliquant l'injéctivité de certains processus croissants, Sém.Proba. XXXII, Lecture Notes in Mathematics 1686, (1998), 316-327. MR1655302
8. J. Azéma, P.A. Meyer, M. Yor: Martingales relatives, Sém.Proba. XXVI, Lecture Notes in Mathematics 1526, (1992), 307-321. MR1232000
9. J. Azéma, M. Yor (eds): Temps locaux, Astérisque 52-53 (1978). MR0509476
10. J. Azéma, M. Yor: Une solution simple au problème de Skorokhod, Sém.Proba. XIII, Lecture Notes in Mathematics 721, (1979), 90-115 and 625-633.
11. J. Azéma, M. Yor: Etude d'une martingale remarquable, Sém.Proba. XXIII, Lecture Notes in Mathematics 1372, (1989), 88-130. MR1022900
12. J. Azéma, M. Yor: Sur les zéros des martingales continues, Sém.Proba. XXVI, Lecture Notes in Mathematics 1526, (1992), 248-306. MR1231999
13. M.T. Barlow, Study of a filtration expanded to include an honest time, ZW, 44, 1978, 307-324. MR0509204
14. M.T. Barlow, J.W. Pitman, M. Yor: Une extension multidimension-nelle de la loi de l'arc sinus, Sém.Proba. XXIII, Lecture Notes in Mathematics 1372, (1989), 294-314.
15. M.T. Barlow, J.W. Pitman, M. Yor: On Walsh's Brownian Motions, Sém.Proba. XXIII, Lecture Notes in Mathematics 1372, (1989), 275-293.
16. M.T. Barlow, M. Yor, Sur la construction d'une martingale continue de valeur absolue donnée, Sém.Proba. XIV, Lecture Notes in Mathematics 784, (1980), 62-75. MR0580109
17. M.T. Barlow, M. Yor: (Semi)-martingale inequalities and local times, Z.W. 55 (1981), 237-254. MR0608019
18. M.T. Barlow, M. Emery, F.B. Knight, S. Song, M. Yor, Autour d'un théorème de Tsirelson sur des filtrations browniennes et non brown-iennes, Sém.Proba. XXXII, Lecture Notes in Mathematics 1686, (1998), 264-305. MR1651223
19. P. Biane, M. Yor, Valeurs principales associées aux temps locaux Brown-iens, Bull. Sci. Maths., 2e série, 111, (1987), 23-101. MR0886959
20. J.M. Bismut, Régularité et continuité des processus, ZW, 44, 1978, 261-268. MR0505956
21. A.N. Borodin, P. Salminen: Handbook of Brownian Motion-Facts and formulae, Birkḧauser. Second edition (2002). MR1912205
22. P. Brémaud, M. Yor: Changes of filtration and of probability measures, Z.f.W, 45, 269-295 (1978). MR0511775
23. K.L. Chung, J.L. Doob, Fields, optionality and measurability, Amer. J. Math, 87, 1965, 397-424. MR0214121
24. C. Dellacherie: Capacités et processus stochastiques, Springer-Verlag (1972). MR0448504
25. C. Dellacherie, P.A. Meyer: A propos du travail de Yor sur les grossissements des tribus, Sém.Proba. XII, Lecture Notes in Mathematics 649, (1978), 69-78 MR0519987
26. C. Dellacherie, P.A. Meyer: Probabilités et potentiel, Hermann, Paris, vol. I 1976.
27. C. Dellacherie, P.A. Meyer: Probabilités et potentiel, Hermann, Paris, volII. 1980. MR0566768
28. C. Dellacherie, B. Maisonneuve, P.A. Meyer: Probabilités et poten-tiel, Chapitres XVII-XXIV: Processus de Markov (fin), Compléments de calcul stochastique, Hermann (1992).
29. E.B. Dynkin: Some limit theorems for sums of independent random vari-ables with infinite mathematical expectations, IMS-AMS Selected Trans. in Math Stat and Prob. 1, 171-189. in preparation. MR0116376
30. R.J. Elliott, M. Jeanblanc, M. Yor: On models of default risk, Math. Finance, 10, 179-196 (2000). MR1802597
31. R.K. Getoor: Excursions of a Markov process, Annals of probability, 7(2), (1979), 244-266. MR0525052
32. D. Gilat: Every nonnegative submartingale is the absolute value of a mar-tingale, Annals of probability, 5, (1977), 475-481. MR0433586
33. P. Imkeller: Random times at which insiders can have free lunches, Stochastics and stochastics reports, 74, (2002), 465-487. MR1940496
34. K. Itô: Poisson point processes attached to Markov Processes, Proc. Sixth. Berkeley Symp. Math. Stat. Prob., vol. 3, (1970), 225-239. MR0402949
35. K. Itô, H.P. McKean: Diffusion processes and their sample paths, Springer (1974). MR0345224
36. K. Itô, S. Watanabe, Transformation of Markov processes by multi-plicative functionals, Ann. Inst. Fourier. Grenoble, 15 1, (1965), 13-30. MR0184282
37. J. Jacod: Calcul stochastique et problèmes de martingales, Lecture Notes in Mathematics 714, (1979), 240-249. MR0542115
38. J. Jacod: Grossissement initial, hypothse (H'), et théorème de Girsanov, In: Grossissements de filtrations: exemples et applications, (ref [45], below) Springer (1985), 15-35. MR0884713
39. J. Jacod, A.N. Shiryaev: Limit theorems for stochastic processes, Springer. Second edition (2003). MR1943877
40. M. Jeanblanc, M. Rutkowski: Modeling default risk: Mathematical tools, Fixed Income and Credit risk modeling and Management, New York University, Stern School of business, Statistics and Operations Research Department, Workshop (2000).
41. T. Jeulin: Grossissement d'une filtration et applications, Sém.Proba. XIII, Lecture Notes in Mathematics 721, (1979), 574-609. MR0544826
42. T. Jeulin: Semi-martingales et grossissements d'une filtration, Lecture Notes in Mathematics 833, Springer (1980). MR0604176
43. T. Jeulin, M. Yor: Grossissement d'une filtration et semimartingales: formules explicites, Sém.Proba. XII, Lecture Notes in Mathematics 649, (1978), 78-97. MR0519998
44. T. Jeulin, M. Yor: Nouveaux résultats sur sur le grossissement des tribus, Ann. Scient. ENS, 4e série, t. 11 (1978), 429-443. MR0521639
45. T. Jeulin, M. Yor (eds): Grossissements de filtrations: exemples et appli-cations, Lecture Notes in Mathematics 1118, Springer (1985). MR0884713
46. F.B. Knight: Local variation of diffusion in local time, Annals of probability, 1, (1973), 1026-1034. MR0397898
47. F.B. Knight: On the sejourn times of killed Brownian Motion, Sém.Proba. XII, Lecture Notes in Mathematics 649, (1978), 428-445. MR0520018
48. F.B. Knight, B. Maisonneuve: A characterization of stopping times. Annals of probability, 22, (1994), 1600-1606. MR1303657
49. P. Lévy: Sur un problème de M. Marcinkiewicz, C.R.A.S, 208, (1939), 318-321. Errata p. 776.
50. R. Mansuy, M. Yor: Random times and (enlargement of filtrations) in a Brownian setting, Lecture Notes in Mathematics, 1873, Springer (2006). MR2200733
51. P.A. Meyer: Processus de Markov, Lecture Notes in Mathematics 26, Springer (1967). MR0219136
52. P.A. Meyer: Le dual de H1 est BMO (cas continu), Sém.Proba. VII, Lecture Notes in Mathematics 321, (1973), 136-145. MR0410910
53. P.A. Meyer: Un cours sur les intégrales stochastiques, Sém.Proba. X, Lecture Notes in Mathematics 511, (1978), 245-400. MR0501332
54. P.A. Meyer: Sur un théorème de J. Jacod, Sém.Proba. XII, Lecture Notes in Mathematics 649, (1978), 57-60. MR0519995
55. P.A. Meyer: Représentations multiplicatives de sousmartingales d'après Azéma, Sém.Proba. XIII, Lecture Notes in Mathematics 721, (1979).
56. P.A. Meyer, C. Yoeurp: Sur la décomposition multiplicative des sous-martingaless positives, Sém.Proba. XIII, Lecture Notes in Mathematics 721, (1979), 240-249.
57. P.W. Millar: Random times and decomposition theorems, Proc. Symp. Pure Math., 31, (1977), 91-103. MR0443109
58. P.W. Millar: A path decomposition for Markov processes, Annals of probability, 6, (1978), 345-348. MR0461678
59. A. Nikeghbali, M. Yor: A definition and some characteristic properties of pseudo-stopping times, Ann. Prob. 33, (2005) 1804-1824. MR2165580
60. A. Nikeghbali: Enlargements of filtrations and path decompositions at non stopping times, Prob. Theory Related Fields, 136 (4), 2006, 524-540.
61. A. Nikeghbali: A class of remarkable submartingales, Stochastic Processes and their Applications 116, (2006) 917-938. MR2254665
62. A. Nikeghbali: Some random times and martingales associated with BES0(δ) processes (0 < δ < 2), Alea 2 67-89 (2006). MR2249663
63. A. Nikeghbali, M. Yor: Doob's maximal identity, multiplicative decom-positions and enlargements of filtrations, Illinois Journal of Mathematics, 50 (4) 791-814 (2006). MR2247846
64. A. Nikeghbali: Non stopping times and stopping theorems, to appear in Stochastic Processes and their Applications
65. A. Nikeghbali: Multiplicative decompositions and frequency of vanishing of nonnegative submartingales, to appear in Journal of Theoretical Probability.
66. A. Nikeghbali: How badly are the Burholder-Davis-Gundy inequalities affected by arbitrary random times?, preprint, available on ArXiv.
67. J.W. Pitman, M. Yor: Bessel processes and infinitely divisible laws, In: D. Williams (ed.) Stochastic integrals, Lecture Notes in Mathematics 851, Springer (1981). MR0620995
68. P.E. Protter: Stochastic integration and differential equations, Springer. Second edition (2005), version 2.1. MR2020294
69. D. Revuz, M. Yor: Continuous martingales and Brownian motion, Springer. Third edition (1999). MR1725357
70. C. Rogers, D. Williams: Diffusions, Markov processes and Martingales, vol 2: Ito calculus, Wiley and Sons, New York, 1987. MR0921238
71. C. Stricker, Quasi-martingales, martingales locales, semimartingales et filtration naturelle, ZW, 39, (1977), 55-63. MR0471072
72. B. Tsirelson: Triple points: from non-Brownian filtrations to harmonic measures, Geom. Func. Anal. 7(6), (1997), 1096-1142. MR1487755
73. D. Williams: Path decomposition and continuity of local time for one-dimensional diffusions I, Proc. London Math. Soc. 3, 28 (1974), 3-28. MR0350881
74. D. Williams: A non stopping time with the optional-stopping property, Bull. London Math. Soc. 34 (2002), 610-612. MR1912883
75. C. Yoeurp: Théorème de Girsanov généralisé, et grossissement d'une fil-tration, In: Grossissements de filtrations: exemples et applications, (ref [45], above) Springer (1985), 172-196. MR0884713
76. M. Yor: A propos d'un lemme de Ch. Yoeurp, Sém.Proba. XI, Lecture Notes in Mathematics 581, (1977), 493-450. MR0451396
77. M. Yor: Grossissements d'une filtration et semi-martingales: théorèmes généraux, Sém.Proba. XII, Lecture Notes in Mathematics 649, (1978), 61-69. MR0519996
78. M. Yor: Les inégalités de sous-martingales comme conséquence de la re-lation de domination, Stochastics 3, (1979), no.1, 1-15. MR0546696
79. M. Yor: Inégalités de martingales continues arr̂ etées 'a un temps quel-conque, I, II, In: Grossissements de filtrations: exemples et applications, (ref [45], above) Springer (1985), 110-171. MR0884713
80. M. Yor: Some aspects of Brownian motion, Part I. Some special function-als. Birkḧauser, Basel (1992). MR1193919
81. M. Yor: Some aspects of Brownian motion, Part II. Some recent martin-gale problems. Birkhauser, Basel (1997). MR1442263