References

1. H. Ahn and P. Protter. Aremark on stochastic integration. In Seminaire de Probabilites XXVIII, volume 1583 of Lecture Notes in Mathematics, pages 312-315. Springer-Verlag, Berlin, 1994.

2. L. Alili, D. Dufresne, and M. Vor. Sur l'identite de Bougerol pour les fonc­tionnelles exponentielles du mouvement brownien avec drift. Biblioteca de la Revista Matematica Iberoamericana, pages 3-14, 1977.

3. L. Arnold. Stochastic Differential Equations: Theory and Applications. Wiley, New York, 1973.

4. M. Atlan, H. Geman, D. Madan, and M. Vor. Correlation and the pricing of risks. Preprint, 2004.

5. J. Azema. Quelques applications de la theorie generale des processus. Invent. Math., 18:293-336, 1972.

6. J. Azema. Sur les fermes aleatoires. In Seminaire de Probabilites XIX, volume 1123 of Lecture Notes in Mathematics, pages 397-495. Springer-Verlag, Berlin, 1985.

7. J. Azema and M. Vor. Etude d'une martingale remarquable. In Seminaire de Probabilites XXIII, volume 1372 of Lecture Notes in Mathematics, pages 88-130. Springer-Verlag, Berlin, 1989.

8. X. Bardina and M. Jolis. An extension of Itö's formula for elliptic diffusion processes. Stochastic Process. Appl., 69:83-109, 1997.

9. M. T. Barlow. Study of a filtration expanded to include an honest time. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 44:307-323, 1978.

10. M. T. Barlow. On the left end points of Brownian excursions. In Seminaire de Probabilites XIII, volume 721 of Lecture Notes in Mathematics, page 646. Springer-Verlag, Berlin, 1979.

11. M. T. Barlow, S. D. Jacka, and M. Vor. Inequalities for a pair of processes stopped at a random time. Proc. London Math. Soc., 52:142-172, 1986.

12. R. F. Bass. Probabilistic Techniques in Analysis. Springer-Verlag, New York, 1995.

13. R. F. Bass. The Doob-Meyer decomposition revisited. Canad. Math. Bull., 39:138-150,1996.

14. J. Bertoin. Levy Processes. Cambridge University Press, Cambridge, 1996. 15. K. Bichteler. Stochastic integrators. Bult. Amer. Math. Soc., 1:761-765, 1979.

398 References

16. K. Bichteler. Stochastic integration and LP-theory of semimartingales. Ann. Probab., 9:49-89, 1981.

17. K. Bichteler. Stochastic Integration With Jumps. Cambridge U niversity Press, Cambridge, 2002.

18. K. Bichteler, J.-B. Gravereaux, and J. Jacod. Malliavin Calculus for Processes With Jumps. Gordon and Breach, New York, 1987.

19. P. Billingsley. Convergence of Probability Measures. Wiley, New York, second edition, 1999.

20. Yu. N. Blagovescenskii and M. I. Freidlin. Certain properties of diffusion processes depending on a parameter. Sov. Math., 2:633-636, 1961. Originally published in Russian in Dokl. Akad. Nauk SSSR, 138:508-511, 1961.

21. R. M. BlumenthaI and R. K. Getoor. Markov Processes and Potential Theory. Academic Press, New York, 1968.

22. T. Bojdecki. Teoria General de Procesos e Integracion Estocastica. Universidad Nacional Autonoma de Mexico, 1988. Segundo edicion.

23. N. Bouleau. Formules de changement de variables. Ann. Inst. H. Poincare Probab. Statist., 20:133-145, 1984.

24. N. Bouleau and M. Vor. Sur la variation quadratique des temps locaux de certaines semimartingales. C. R. Acad. Sei. Paris Sero I Math., 292:491-494, 1981.

25. L. Breiman. Probability. SIAM, Philadelphia, 1992. 26. P. Bremaud. Point Processes and Queues: Martingale Dynamies. Springer­

Verlag, New York, 1981. 27. P. Bremaud and M. Vor. Changes of filtrations and probability measures. Z.

Wahrscheinlichkeitstheorie verw. Gebiete, 45:269-295, 1978. 28. J. L. Bretagnolle. Processus a accroissements independants. In Ecole d'Ete de

Probabilites: Processus Stochastiques., volume 307 of Lecture Notes in Mathe­matics, pages 1-26. Springer-Verlag, Berlin, 1973.

29. B. Bru and M. Vor. Comments on the life and mathematicallegacy ofWolfgang Doeblin. Finance Stoch., 6:3-47, 2002.

30. M. Chaleyat-Maurel and Th. Jeulin. Grossissement gaussien de la filtration brownienne. In Grossissements de filtrations: exemples et applications, volume 1118 of Lecture Notes in Mathematics, pages 59-109. Springer-Verlag, Berlin, 1985.

31. C. S. Chou. Caracterisation d'une classe de semimartingales. In Seminaire de Probabilites XIII, volume 721 of Lecture Notes in Mathematics, pages 250-252. Springer-Verlag, Berlin, 1979.

32. C. S. Chou and P. A. Meyer. Sur la representation des martingales comme integrales stochastiques dans les processus ponctuels. In Seminaire de Prob­abilites IX, volume 465 of Lecture Notes in Mathematics, pages 226-236. Springer-Verlag, Berlin, 1975.

33. C. S. Chou, P. A. Meyer, and C. Stricker. Sur les integrales stochastiques de processus previsibles non bornes. In Seminaire de Probabilites XIV, volume 784 of Lecture Notes in Mathematics, pages 128-139. Springer-Verlag, Berlin, 1980.

34. K. L. Chung and R. J. Williams. Introduction to Stochastic Integration. Birkhäuser, Boston, second edition, 1990.

35. E. Qinlar. Introduction to Stochastic Processes. Prentice Hall, Englewood Cliffs, 1975.

References 399

36. E. Qinlar, J. Jacod, P. Protter, and M. Sharpe. Semimartingales and Markov processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 54: 161 ~220, 1980.

37. E. A. Coddington and N. Levinson. Theory 0/ Ordinary Differential Equations. McGraw-Hill, New York, 1955.

38. P. Courrege. Integrale stochastique par rapport EI une martingale de carre integrable. In Seminaire de Theorie du Potentiel dirige par M. Brelot, G. Cho­quet et J. Deny, 7e annee {1962/1963}, Paris, 1963. Faculte des Sciences de Paris.

39. K. E. Dambis. On the decomposition of continuous submartingales. Theory Probab. Appl., 1O:401~410, 1965.

40. F. Delbaen and W. Schachermayer. The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann., 312:215~250, 1998.

41. F. Delbaen and M. Vor. Passport options. Math. Finance, 12:299~328, 2002. 42. C. Dellacherie. Un exemple de la theorie generale des processus. In Seminaire

de Probabilites IV, volume 124 of Lecture Notes in Mathematics, pages 60~70. Springer-Verlag, Berlin, 1970.

43. C. Dellacherie. Capacites et processus stochastiques. Springer-Verlag, Berlin, 1972.

44. C. Dellacherie. Un survol de la theorie de l'integrale stochastique. Stochastic Process. Appl., 10:115~144, 1980.

45. C. Dellacherie. Mesurabilite des debuts et theoreme de section. In Seminaire de Probabilites XV, volume 850 of Lecture Notes in Mathematics, pages 351~360. Springer-Verlag, Berlin, 1981.

46. C. Dellacherie, B. Maisonneuve, and P. A. Meyer. Probabilites et Potentiel. Chapitres XVII a XXIV; Processus de Markov (fin); Complements de Calcul Stochastique. Hermann, Paris, 1992.

47. C. Dellacherie and P. A. Meyer. Probabilities and Potential, volume 29 of North-Holland Mathematics Studies. North-Holland, Amsterdam, 1978.

48. C. Dellacherie and P. A. Meyer. Probabilities and Potential B, volume 72 of North-Holland Mathematics Studies. North-Holland, Amsterdam, 1982.

49. C. Dellacherie, P. A. Meyer, and M. Vor. Sur certaines proprietes des espaces de Banach 1{1 et EMO. In Seminaire de Probabilites XII, volume 649 of Lecture Notes in Mathematics, pages 98~113. Springer-Verlag, Berlin, 1978.

50. C. Doleans-Dade. Integrales stochastiques dependant d'un parametre. Publ. Inst. Stat. Univ. Paris, 16:23~34, 1967.

51. C. Doleans-Dade. Quelques applications de la formule de changement de vari­ables pour les semimartingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 16:181~194, 1970.

52. C. Doleans-Dade. Une martingale uniformement integrable mais non locale­ment de carre integrable. In Seminaire de Probabilites V, volume 191 of Lecture Notes in Mathematics, pages 138~140. Springer-Verlag, Berlin, 1971.

53. C. Doleans-Dade. On the existence and unicity of solutions of stochastic dif­ferential equations. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 36:93~101, 1976.

54. C. Doleans-Dade. Stochastic processes and stochastic differential equations. In Stochastic Differential Equations: I ciclo 1978, Palazzone della Scuola Normale Superiore, Cortona (Centro Internazionale Matematico Estivo), pages 5~75. Liguori Editore, Napoli, 1981.

400 References

55. C. Doleans-Dade and P. A. Meyer. Integrales stochastiques par rapport aux martingales locales. In Seminaire de Pmbabilites IV, volume 124 of Lecture Notes in Mathematics, pages 77-107. Springer-Verlag, Berlin, 1970.

56. C. Doleans-Dade and P. A. Meyer. Equations differentielles stochastiques. In Seminaire de Pmbabilites XI, volume 581 of Lecture Notes in Mathematics, pages 376-382. Springer-Verlag, Berlin, 1977.

57. J. L. Doob. Stochastic Pmcesses. Wiley, New York, 1953. 58. J. L. Doob. Classical Potential Theory and Its Pmbabilistic Counterpart.

Springer-Verlag, Berlin, 1984. 59. H. Doss. Liens entre equations differentielles stochastiques et ordinaires. Ann.

Inst. H. Poincani Pmbab. Statist., 13:99-125, 1977. 60. H. Doss and E. Lenglart. Sur l'existence, l'unicite et le comportement asymp­

totique des solutions d'equations differentielles stochastiques. Ann. Inst. H. Poincare Pmbab. Statist., 14:189-214, 1978.

61. L. Dubins and G. Schwarz. On continuous martingales. Pmc. Natl. Acad. Sci. USA, 53:913-916, 1965.

62. D. Duffie and C. Huang. Multiperiod security markets with differential infor­mation. J. Math. Econom., 15:283-303, 1986.

63. D. A. Edwards. A note on stochastic integrators. Math. Pmc. Cambridge Philos. Soc., 107:395-400, 1990.

64. N. Eisenbaum. Integration with respect to local time. Potential Anal., 13:303-328,2000.

65. Y. El-Khatib and N. Privault. Hedging in complete markets driven by normal martingales. Appl. Math. (Warsaw), 30:147-172, 2003.

66. R. J. Elliott. Stochastic Calculus and Applications. Springer-Verlag, Berlin, 1982.

67. M. Emery. Stabilite des solutions des equations differentielles stochastiquesj applications aux integrales multiplicatives stochastiques. Z. Wahrscheinlich­keitstheorie verw. Gebiete, 41:241-262, 1978.

68. M. Emery. Equations differentielles stochastiques lipschitziennes: etude de la stabilite. In Seminaire de Pmbabilites XIII, volume 721 of Lecture Notes in Mathematics, pages 281-293. Springer-Verlag, Berlin, 1979.

69. M. Emery. Une topologie sur l'espace des semimartingales. In Seminaire de Pmbabilites XIII, volume 721 of Lecture Notes in Mathematics, pages 260-280. Springer-Verlag, Berlin, 1979.

70. M. Emery. Compensation de processus a variation finie non localement integrables. In Seminaire de Pmbabilites XIV, volume 784 of Lecture Notes in Mathematics, pages 152-160. Springer-Verlag, Berlin, 1980.

71. M. Emery. Non confluence des solutions d'une equation stochastique lipschitzi­enne. In Seminaire de Pmbabilites XV, volume 850 of Lecture Notes in Math­ematics, pages 587-589. Springer-Verlag, Berlin, 1981.

72. M. Emery. On the Azema martingales. In Seminaire de Pmbabilites XXIII, volume 1372 of Lecture Notes in Mathematics, pages 66-87. Springer-Verlag, Berlin, 1989.

73. M. Emery and M. Vor, editors. Seminaire de Pmbabilites 1967-1980: A Se­lection in Martingale Theory, volume 1771 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002.

74. S. Ethier and T. G. Kurtz. Markov Pmcesses: Chamcterization and Conver­gence. Wiley, New York, 1986.

References 401

75. W. Feller. An Introduetion to Probability Theory and Its Applieations. Volume II. Wiley, New York, second edition, 1971.

76. D. L. Fisk. Quasimartingales. Trans. Amer. Math. Soe., 120:369-389, 1965. 77. T. R. Fleming and D. P. Harrington. Counting Proeesses and Survival Analysis.

John Wiley, New York, 1991. 78. H. Föllmer. Calcul d'Itö sans probabilites. In Seminaire de Probabilites XV,

volume 850 of Leeture Notes in Mathematies, pages 144-150. Springer-Verlag, Berlin, 1981.

79. H. Föllmer and P. Protter. On Itö's formula for multidimensional Brownian motion. Probab. Theory Related Fields, 116:1-20,2000.

80. H. Föllmer, P. Protter, and A. N. Shiryayev. Quadratic covariation and an extension of Itö's formula. Bernoulli, 1:149-169, 1995.

81. T. Fujiwara and H. Kunita. Stochastic differential equations of jump type and Levy processes in diffeomorphisms group. J. Math. Kyoto Univ., 25:71-106, 1985.

82. R. K. Getoor and M. Sharpe. Conformal martingales. Invent. Math., 16:271-308, 1972.

83. R. Ghomrasni and G. Peskir. Loeal time-space calculus and extensions of Itö's formula. In High Dimensional Probability III (Sandbjerg, Denmark 2002), volume 55 of Progress in Probability, pages 177-192. Birkhäuser Basel, 2003.

84. I. I. Gihman and A. V. Skorohod. Stoehastie Differential Equations. Springer­Verlag, Berlin, 1972.

85. R. D. Gill and S. Johansen. A survey of product-integration with a view towards applications in survival analysis. Ann. Statist., 18:1501-1555, 1990.

86. I. V. Girsanov. On transforming a certain dass of stoehastic processes by absolutely continuous substitutions of measures. Theory Probab. Appl., 5:285-301, 1960.

87. M. Greenberg. Lectures on Algebraie Topology. Benjamin, New York, 1967. 88. T. H. Gronwall. Note on the derivatives with respect to a parameter of the

solutions of a system of differential equations. Ann. 0/ Math. (2), 20:292-296, 1919.

89. A. Gut. An introduction to the theory of asymptotic martingales. In Amarts and Set Function Proeesses, volume 1042 of Lecture Notes in Mathematies, pages 1-49. Springer-Verlag, Berlin, 1983.

90. S. W. He, J. G. Wang, and J. A. Yan. Semimaningale Theory and Stoehastie Caleulus. CRC Press, Boca Raton, 1992.

91. T. Hida. Brownian Motion. Springer-Verlag, Berlin, 1980. 92. G. A. Hunt. Markoff processes and potentials I. Illinois J. Math., 1:44-93,

1957. 93. G. A. Hunt. Markoff processes and potentials II. Illinois J. Math., 1:316-369,

1957. 94. G. A. Hunt. Markoff processes and potentials IH. Illinois J. Math., 2:151-213,

1958. 95. N. Ikeda and S. Watanabe. Stoehastie Differential Equations and Diffusion

Proeesses. North-Holland, Amsterdam, 1981. 96. D. Isaacson. Stochastic integrals and derivatives. Ann. Math. Sei., 40:1610-

1616, 1969. 97. K. Itö. Stochastic integral. Proc. Imp. Aead. Tokyo, 20:519-524, 1944. 98. K. Itö. On stochastic integral equations. Proe. Japan Aead., 22:32-35, 1946.

402 References

99. K. Ito. Stochastic differential equations in a differentiable manifold. Nagoya Math. J., 1:35-47, 1950.

100. K. Ito. Multiple Wiener integral. J. Math. Soc. Japan, 3:157-169, 1951. 101. K. Ito. On a formula concerning stochastic differentials. Nagoya Math. J.,

3:55-65, 1951. 102. K. Ito. On Stochastic Differential Equations. Memoirs of the American Math­

ematical Society, 4, 1951. 103. K. Ito. Stochastic differentials. Appl. Math. Optim., 1:374-381, 1974. 104. K. Ito. Extension of stochastic integrals. In Proceedings of International Sym­

posium on Stochastic Differential Equations, pages 95-109, New York, 1978. Wiley.

105. K. Ito and H. P. McKean Jr. Diffusion Processes and Their Sampie Paths. Springer-Verlag, Berlin, 1965; Reissued in paperback, 2002.

106. K. Ito and M. Nisio. On stationary solutions of a stochastic differential equa­tion. J. Math. Kyoto Univ., 4:1-75, 1964.

107. K. Ito and S. Watanabe. Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier, 15:15-30, 1965.

108. J. Jacod. Calcul Stochastique et Problemes de Martingales, volume 714 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1979.

109. J. Jacod. Grossissement initial, hypothese (H'), et theoreme de Girsanov. In Grossissements de jiltmtions: exemples et applications, volume 1118 of Lecture Notes in Mathematics, pages 15-35. Springer-Verlag, Berlin, 1985.

110. J. Jacod, T. G. Kurtz, S. Meleard, and P. Protter. The approximate Euler method for Levy driven stochastic differential equations. To appear in Ann. Inst. H. Poincare Probab. Statist., 2004.

111. J. Jacod, S. Meleard, and P. Protter. Explicit form and robustness of martin­gale representations. Ann. Probab., 28:1747-1780, 2000.

112. J. Jacod and P. Protter. Time revers al on Levy processes. Ann. Probab., 16:620-641, 1988.

113. J. Jacod and P. Protter. Une remarque sur les equations differentielles stochas­tiques a solutions Markoviennes. In Seminaire de Probabilites XXV, volume 1485 of Lecture Notes in Mathematics, pages 138-139. Springer-Verlag, Berlin, 1991.

114. J. Jacod and P. Protter. Probability Essentials. Springer-Verlag, Heidelberg, second edition, 2003.

115. J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Springer-Verlag, New York, second edition, 2003.

116. J. Jacod and M. Yor. Etude des solutions extremales et representation integrale des solutions pour certains problemes de martingales. Z. Wahrscheinlichkeits­theorie veTW. Gebiete, 38:83-125, 1977.

117. S. Janson and M. J. Wichura. Invariance principles for stochastic area and related stochastic integrals. Stochastic Process. Appl., 16:71-84, 1983.

118. R. Jarrow and P. Protter. A short history of stochastic integration and math­ematical finance: The early years, 1880-1970. In A Festschrift for HeTman Rubin, volume 45 of IMS Lecture Notes Monogmph, pages 75-91. Institue of Mathematical Statistics, Beachwood, Ohio, 2004.

119. Th. Jeulin. Semi-Martingales et Grossissement d'une Filtmtion, volume 833 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1980.

Referenees 403

120. Th. Jeulin. Sur la eonvergenee absolue de eertaines integrales. In Seminaire de Probabilites XVI, volume 920 of Leeture Notes in Mathematics, pages 248-256. Springer-Verlag, Berlin, 1982.

121. Th. Jeulin and M. Yor. Grossissement d'une filtration et semi-martingales: formules explieites. In Seminaire de Probabilites XII, volume 649 of Lecture Notes in Mathematics, pages 78-97. Springer-Verlag, Berlin, 1978.

122. Th. Jeulin and M. Yor. Nouveaux resultats sur le grossissement des tribus. Ann. Scient. E.N.S., 4e serie, 11:429-443, 1978.

123. Th. Jeulin and M. Yor. Inegalite de Hardy, semimartingales, et faux-amis. In Seminaire de Probabilites XIII, volume 721 of Lecture Notes in Mathematics, pages 332-359. Springer-Verlag, Berlin, 1979.

124. Th. Jeulin and M. Yor, editors. Grossissements de jiltrations: exemples et applications, volume 1118 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1985.

125. Th. Jeulin and M. Yor. Reeapitulatif. In Grossissements de jiltrations: exemples et applications, volume 1118 of Lecture Notes in Mathematics, pages 305-315. Springer-Verlag, Berlin, 1985.

126. G. Johnson and L. L. Helms. Class D supermartingales. Bull. Amer. Math. Soc., 69:59-62, 1963.

127. T. Kailath, A. Segall, and M. Zakai. Fubini-type theorems for stoehastic inte­grals. Sankhyii Sero A, 40:138-143, 1978.

128. G. Kallianpur and C. Striebel. Stoehastic differential equations oeeuring in the estimation of eontinuous parameter stoehastic proeesses. Theory Probab. Appl., 14:567-594, 1969.

129. I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus. Springer­Verlag, New York, seeond edition, 1991.

130. A. F. Karr. Point Processes and Their Statistical Inference. Mareel Dekker, New York, seeond edition, 1991.

131. N. Kazamaki. Note on a stoehastic integral equation. In Seminaire de Probabilites VI, volume 258 of Lecture Notes in Mathematics, pages 105-108. Springer-Verlag, Berlin, 1972.

132. N. Kazamaki. On a problem of Girsanov. T8hoku Math. J. (2), 29:597-600, 1977.

133. N. Kazamaki. Continuous Exponential Martingales and BMO, volume 1579 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1994.

134. J. F. C. Kingman. An intrinsie deseription of loeal time. J. London Math. Soc., 6:725-731, 1973.

135. J. F. C. Kingman and S. J. Taylor. Introduction to Measure and Probability. Cambridge University Press, Cambridge, 1966.

136. P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin, 1992.

137. F. B. Knight. Calculating the eompensator: method and example. In Seminar on Stochastic Processes, 1990, pages 241-252, Boston, 1991. Birkhäuser.

138. P. E. Kopp. Martingales and Stochastic Integrals. Cambridge University Press, Cambridge, 1984.

139. H. Kunita. On the deeomposition of solutions of stoehastie differential equa­tions. In Stochastic Integrals, volume 851 of Lecture Notes in Mathematics, pages 213-255. Springer-Verlag, Berlin, 1981.

404 References

140. H. Kunita. Some extensions of Itö's formula. In Seminaire de Probabilites XV, volume 850 of Lecture Notes in Mathematics, pages 118-141. Springer-Verlag, Berlin, 1981.

141. H. Kunita. Stochastic differential equations and stochastic flow of diffeomor­phisms. In Ecole d'Ete de Probabilites de Saint-Flour XII-1982, volume 1097 of Lecture Notes in Mathematics, pages 143-303. Springer-Verlag, Berlin, 1984.

142. H. Kunita. Lectures on stochastic fiows and applications, volume 78 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Springer-Verlag, Berlin, 1986.

143. H. Kunita and S. Watanabe. On square integrable martingales. Nagoya Math. J., 30:209-245, 1967.

144. T. G. Kurtz. Lectures on Stochastic Analysis. Unpublished, 2001. Available online as of July 2004 at http://www.math.wisc.edu/rvkurtz.

145. T. G. Kurtz and P. Protter. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab., 19:1035-1070, 1991.

146. T. G. Kurtz and P. Protter. Wong-Zakai corrections, random evolutions, and simulation schemes for SDEs. In Stochastic Analysis: Liber Amicorum for Moshe Zakai, pages 331-346. Academic Press, 1991.

147. A. U. Kussmaul. Stochastic Integration and Generalized Martingales, volume 11 of Pitman Research Notes in Mathematics. Longman Scientific & Technical, Guildford, 1977.

148. G. Last and A. Brandt. Marked Point Processes on the Real Line. Springer­Verlag, New York, 1995.

149. J.F. Le GaU. Applications du temps local aux equations differentielles stochas­tiques unidimensionnelles. In Seminaire de Probabilites XVII, volume 986 of Lecture Notes in Mathematics, pages 15-31. Springer-Verlag, Berlin, 1983.

150. Y. Le Jan. Martingales et changement de tempsj martingales de sauts. C. R. Acad. Sei. Paris Sero A-B, 284:A1151-A1153, 1977.

151. Y. Le Jan. Temps d'arret stricts et martingales de sauts. Z. Wahrscheinlich­keitstheorie verw. Gebiete, 44:213-225, 1978.

152. Y. Le Jan. Martingales et changements de temps. In Seminaire de Probabilites XIII, volume 721 of Lecture Notes in Mathematics, pages 385-399. Springer­Verlag, Berlin, 1979.

153. R. Leandre. Un exemple en theorie des flots stochastiques. In Seminaire de Probabilites XVII, volume 986 of Lecture Notes in Mathematics, pages 158-161. Springer-Verlag, Berlin, 1983.

154. R. Leandre. Flot d'une equation differentielle stochastique avec semimartin­gale directrice discontinue. In Seminaire de Probabilites XIX, volume 1123 of Lecture Notes in Mathematics, pages 271-274. Springer-Verlag, Berlin, 1985.

155. E. Lenglart. Transformation des martingales locales par changement absol­ument continu de ·probabilites. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 39:65-70, 1977.

156. E. Lenglart. Une caracterisation des processus previsibles. In Seminaire de Probabilites XI, volume 581 of Lecture Notes in Mathematics, pages 415-417. Springer-Verlag, Berlin, 1977.

157. E. Lenglart. Appendice a l'expose precedent: inegalites de semimartingales. In Seminaire de Probabilites XIV, volume 784 of Lecture Notes in Mathematics, pages 49-52. Springer-Verlag, Berlin, 1980.

158. E. Lenglart. Semi-martingales et integrales stochastiques en temps continu. Rev. CETHEDEC, 75:91-160,1983.

References 405

159. E. Lenglart, D. Lepingle, and M. Pratelli. Presentation unifiee de certaines inegalites de la theorie des martingales. In Seminaire de Pmbabilites XIV, volume 784 of Lecture Notes in Mathematics, pages 26~48. Springer-Verlag, Berlin, 1980.

160. D. Lepingle. Une remarque sur les lois de certains temps d'atteinte. In Seminaire de Pmbabilites XV, volume 850 of Lecture Notes in Mathematics, pages 669~670. Springer-Verlag, Berlin, 1981.

161. D. Lepingle and J. Memin. Sur l'integrabilite uniforme des martingales expo­nentielles. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 42:175~203, 1978.

162. G. Letta. Martingales et integration stochastique. Scuola Normale Superiore, Pisa, 1984.

163. P. Levy. Pmcessus stochastiques et mouvement Bmwnien. Gauthier-Villars, Paris, 1948.

164. P. Levy. Wiener's random function, and other Laplacian random functions. In Pmceedings oi the Second Berkeley Symposium on Mathematical Statistics and Pmbability, pages 171~ 187, Berkeley, 1951. University of California Press.

165. T. Lindvall. W. Doeblin 1915~1940. Ann. Probab., 19:929~934, 1991. 166. D. Madan and M. Vor. !tö's integrated formula for strict local martingales.

Preprint, 2004. 167. B. Maisonneuve. Une mise au point sur les martingales continues definies sur

un intervalle stochastique. In Seminaire de Pmbabilites XI, volume 581 of Lecture Notes in Mathematics, pages 435-445. Springer-Verlag, Berlin, 1977.

168. G. Maruyama. On the transition prob ability functions of the Markov process. Nat. Sci. Rep. Ochanomizu Univ., 5:1O~20, 1954.

169. H. P. McKean. Stochastic Integrals. Academic Press, New York, 1969. 170. E. J. McShane. Stochastic Calculus and Stochastic Models. Academic Press,

New York, 1974. 171. E. J. McShane. Stochastic differential equations. J. Multivariate Anal., 5:121-

177, 1975. 172. J. Memin. Decompositions multiplicatives de semimartingales exponentielles

et applications. In Seminaire de Probabilites XII, volume 649 of Lecture Notes in Mathematics, pages 35~46. Springer-Verlag, Berlin, 1978.

173. J. Memin and A. N. Shiryaev. Un critere previsible pour l'uniforme integrabilite des semimartingales exponentielles. In Seminaire de Pmbabilites XIII, volume 721 of Lecture Notes in Mathematics, pages 147-161. Springer-Verlag, Berlin, 1979.

174. M. Metivier. Semimartingales: A Course on Stochastic Pmcesses. Walter de Gruyter, Berlin and New York, 1982.

175. M. Metivier and J. Pellaumail. On Doleans-Föllmer's measure for quasimartin­gales. Illinois J. Math., 19:491~504, 1975.

176. M. Metivier and J. Pellaumail. Mesures stochastiques a valeurs dans les espaces Lo. Z. Wahrscheinlichkeitstheorie veTW. Gebiete, 40:101~114, 1977.

177. M. Metivier and J. Pellaumail. On a stopped Doob's inequality and general stochastic equations. Ann. Pmbab., 8:96-114, 1980.

178. M. Metivier and J. Pellaumail. Stochastic Integration. Academic Press, New Yark,1980.

179. P. A. Meyer. A decomposition theorem far supermartingales. Illinois J. Math., 6:193-205, 1962.

180. P. A. Meyer. Decomposition of supermartingales: the uniqueness theorem. Illinois J. Math., 7:1~17, 1963.

406 Referenees

181. P. A. Meyer. Probability and Potentials. Blaisdell, Waltham, 1966. 182. P. A. Meyer. Integrales stoehastiques I. In Seminaire de Probabilites I, vol­

urne 39 of Lecture Notes in Mathematics, pages 72-94. Springer-Verlag, Berlin, 1967.

183. P. A. Meyer. Integrales stoehastiques 11. In Seminaire de Probabilites I, vol­urne 39 of Lecture Notes in Mathematics, pages 95-117. Springer-Verlag, Berlin, 1967.

184. P. A. Meyer. Integrales stoehastiques III. In Seminaire de Probabilites I, volume 39 of Lecture Notes in Mathematics, pages 118-141. Springer-Verlag, Berlin, 1967.

185. P. A. Meyer. Integrales stoehastiques IV. In Seminaire de Probabilites I, volume 39 of Lecture Notes in Mathematics, pages 142-162. Springer-Verlag, Berlin, 1967.

186. P. A. Meyer. Le dual de "H1 " est "BMO" (eas eontinu). In Seminaire de Probabilites VII, volume 3210f Lecture Notes in Mathematics, pages 136-145. Springer-Verlag, Berlin, 1973.

187. P. A. Meyer. Un cours sur les integrales stoehastiques. In Seminaire de Proba­bilites X, volume 511 of Lecture Notes in Mathematics, pages 246-400. Springer­Verlag, Berlin, 1976.

188. P. A. Meyer. Le theoreme fondamental sur les martingales loeales. In Seminaire de Probabilites XI, volume 581 of Lecture Notes in Mathematics, pages 463-464. Springer-Verlag, Berlin, 1977.

189. P. A. Meyer. Sur un theoreme de C. Stricker. In Seminaire de Probabilites XI, volume 581 of Lecture Notes in Mathematics, pages 482-489. Springer-Verlag, Berlin, 1977.

190. P. A. Meyer. Inegalites de normes pour les integrales stoehastiques. In Seminaire de Probabilites XII, volume 649 of Lecture Notes in Mathematics, pages 757-762. Springer-Verlag, Berlin, 1978.

191. P. A. Meyer. Sur un theoreme de J. Jaeod. In Seminaire de Probabilites XII, volume 649 of Lecture Notes in Mathematics, pages 57-60. Springer-Verlag, Berlin, 1978.

192. P. A. Meyer. Caraeterisation des semimartingales, d'apres Dellaeherie. In Seminaire de Probabilites XIII, volume 721 of Lecture Notes in Mathematics, pages 620-623. Springer-Verlag, Berlin, 1979.

193. P. A. Meyer. Flot d'une equation differentielle stoehastique. In Seminaire de Probabilites XV, volume 850 of Lecture Notes in Mathematics, pages 103-117. Springer-Verlag, Berlin, 1981.

194. P. A. Meyer. Geometrie differentielle stoehastique. In Colloque en l'honneur de Laurent Schwartz (Ecole Polytechnique, 30 mai-3 juin 1983). Vol. 1, volume 131 of Asterisque, pages 107-114. Societe Mathematique de Franee, Paris, 1985.

195. P. A. Meyer. Construetion de solutions d'equation de strueture. In Seminaire de Probabilites XXIII, volume 1372 of Lecture Notes in Mathematics, pages 142-145. Springer-Verlag, Berlin, 1989.

196. T. Mikoseh. Elementary Stochastic Calculus with Finance in View. World Scientifie, Singapore, 1998.

197. P. W. Miliar. Stoehastic integrals and proeesses with stationary independent inerements. In Proceedings 0/ the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Vol. III: Probability theory, pages 307-331, Berkeley, 1972. University of California Press.

References 407

198. D. S. Mitrinovic, J. E. PecariC, and A. M. Fink. Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer, Dordrecht, 1991.

199. J. R. Munkries. Topology: A First Course. Prentice-Hall, Englewood Cliffs, 1975.

200. A. A. Novikov. On an identity for stochastic integrals. Theory Probab. Appl., 17:717-720, 1972.

201. C. O'Cinneide and P. Protter. An elementary approach to naturality, pre­dictability, and the fundamental theorem of local martingales. Stoch. Models, 17:449-458, 2001.

202. B. 0ksendal. Stochastic Differential Equations: An Introduction with Applica­tions. Springer-Verlag, New York, sixth edition, 2003.

203. S. Orey. F-processes. In Proceedings 0/ the Fifth Berkeley Symposium on Mathematical Statistics and Probability. Vol. II: Contributions to probability theory. Part 1, pages 301-313, Berkeley, 1965. University of California Press.

204. y. Ouknine. Generalisation d'un lemme de S. Nakao et applications. Stochas­tics, 23:149-157, 1988.

205. Y. Ouknine. Temps local du produit et du sup de deux semimartingales. In Seminaire de Probabilites XXIV, volume 1426 of Lecture Notes in Mathematics, pages 477-479. Springer-Verlag, Berlin, 1990.

206. E. Pardoux. Grossissement d'une filtration et retournement du temps d'une diffusion. In Seminaire de Probabilites XX, volume 1204 of Lecture Notes in Mathematics, pages 48-55. Springer-Verlag, Berlin, 1986.

207. E. Pardoux. Book Review of Stochastic Integration and Differential Equations, first edition, by P. Protter. Stoch. Stoch. Rep., 42:237-240, 1993.

208. J. Pellaumail. Sur l'integrale stochastique et la decomposition de Doob-Meyer, volume 9 of Asterisque. Societe MatMmatique de France, Paris, 1973.

209. J. Picard. Une classe de processus stable par retournement du temps. In Seminaire de Probabilites XX, volume 1204 of Lecture Notes in Mathematics, pages 56-67. Springer-Verlag, Berlin, 1986.

210. Z. R. Pop-Stojanovic. On McShane's belated stochastic integral. SIAM J. Appl. Math., 22:87-92, 1972.

211. M. H. Protter and C. B. Morrey. A First Course in Real Analysis. Springer­Verlag, Berlin, 1977.

212. P. Protter. Markov solutions of stochastic differential equations. Z. Wahr­scheinlichkeitstheorie verw. Gebiete, 41:39-58, 1977.

213. P. Protter. On the existence, uniqueness, convergence, and explosions of so­lutions of systems of stochastic integral equations. Ann. Probab., 5:243-261, 1977.

214. P. Protter. Right-continuous solutions of systems of stochastic integral equa­tions. J. Multivariate Anal., 7:204-214, 1977.

215. P. Protter. HP stability of solutions of stochastic differential equations. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 44:337-352, 1978.

216. P. Protter. A comparison of stochastic integrals. Ann. Probab., 7:176-189, 1979.

217. P. Protter. Semimartingales and stochastic differential equations. Technical Report 85-25, Purdue University Department of Statistics, 1985.

218. P. Protter. Stochastic integration without tears. Stochastics, 16:295-325, 1986. 219. P. Protter. A partial introduction to financial asset pricing theory. Stochastic

Process. Appl., 91:169-203, 2001.

408 References

220. P. Protter and M. Sharpe. Martingales with given absolute value. Ann. Probab., 7: 1056-1058, 1979.

221. P. Protter and D. Talay. The Euler scheme for Levy driven stochastic differ­ential equations. Ann. Probab., 25:393-423, 1997.

222. K. M. Rao. On decomposition theorems of Meyer. Math. Scand., 24:66-78, 1969.

223. K. M. Rao. Quasimartingales. Math. Scand., 24:79-92, 1969. 224. D. Revuz and M. Yor. Continuous Martingales and Brownian Motion.

Springer-Verlag, New York, third edition, 1999. 225. L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martin­

gales. Volume 1: Foundations. Cambridge University Press, Cambridge, second edition, 2000.

226. L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martin­gales. Volume 2: It8 Calculus. Cambridge University Press, Cambridge, second edition, 2000.

227. H. Rootzen. Limit distributions for the error in approximations of stochastic integrals. Ann. Probab., 8:241-251, 1980.

228. F. Russo and P. Vallois. Itö formula for C1-functions of semimartingales. Probab. Theory and Related Fields, 104:27-41, 1996.

229. G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York, 1994.

230. J.H. Van Schuppen and E. Wong. Transformation of local martingales under a change of law. Ann. Probab., 2:879-888, 1974.

231. L. Schwartz. Semimartingales and their stochastic calculus on manifolds. Presses de l'Universite de Montreal, Montreal, 1984.

232. L. Schwartz. Convergence de la serie de Picard pour les EDS. In Seminaire de Probabilites XXIII, volume 1372 of Lecture Notes in Mathematics, pages 343-354. Springer-Verlag, Berlin, 1989.

233. M. Sharpe. General Theory of Markov Processes. Academic Press, New York, 1988.

234. K. Shimbo. A Novikov type condition for martingales with jumps. Ph.D. dissertation, Cornell University, Ithaca, NY, 2005.

235. J. M. Steele. Stochastic Calculus and Financial Applications. Springer-Verlag, New York, 2001. Corrected third printing, 2003.

236. R. L. Stratonovich. A new representation for stochastic integrals. SIAM J. Control, 4:362-371, 1966.

237. C. Stricker. Quasimartingales, martingales locales, semimartingales, et filtra­tions naturelles. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 3:55-64, 1977.

238. C. Stricker and M. Yor. Calcul stochastique dependant d'un paramEMe. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 45:109-134, 1978.

239. D. W. Stroock. On the growth of stochastic integrals. Z. Wahrscheinlichkeits­theorie verw. Gebiete, 18:340-344, 1971.

240. D. W. Stroock. Markov Processes !rom K. It8 's Perspective. Princeton Uni­versity Press, Princeton, 2003.

241. D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Springer-Verlag, Berlin, 1979.

242. P. Sundar. Time reversal of solutions of equations driven by Levy processes. In Diffusion processes and related problems in analysis, Vol. 11, volume 27 of Progress in Probability, pages 111-119. Birkhäuser Boston, 1992.

Referenees 409

243. H. Sussman. On the gap between deterministic and stoehastic ordinary differ­ential equations. Ann. Probab., 6:19-41, 1978.

244. A. S. Sznitman. Martingales dependant d'un parametre: une formule d'Itö. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 60:41-70, 1982.

245. F. Treves. Topological Vector Spaces, Distributions and Kemels. Aeademic Press, New York, 1967.

246. A. Uppman. Deux applieations des semi-martingales exponentielles aux equations differentielles stoehastiques. C. R. Acad. Sei. Paris Sero A-B, 290:A661-A664, 1980.

247. A. Uppman. Sur le flot d'une equation differentielle stoehastique. In Seminaire de Probabilites XVI, volume 920 of Lecture Notes in Mathematies, pages 268-284. Springer-Verlag, Berlin, 1982.

248. H. von Weizsäeker and G. Winkler. Stochastic Integmls. Vieweg, Braunschweig, 1990.

249. J. Walsh. A diffusion with a diseontinuous loeal time. In Temps locaux: Exposes du seminaire J. Azema, M. Yor (1976-1977), volume 52-53 of Asterisque, pages 37-45. Societe Mathematique de Franee, Paris, 1978.

250. A. T. Wang. Generalized Itö's formula and additive functionals ofthe Brownian path. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 41:153-159, 1977.

251. N. Wiener. Differential space. J. Math. Phys., 2:131-174,1923. 252. N. Wiener. The Dirichlet problem. J. Math. Phys., 3:127-146, 1924. 253. N. Wiener. The quadratic variation of a function and its Fourier eoefficients.

J. Math. Phys., 3:72-94, 1924. 254. E. Wong. Stochastic Processes in Information and Dynamical Systems. Me­

Graw Hill, New York, 1971. 255. T. Yamada. Sur une construetion des solutions d'equations differentielles sto­

ehastiques dans le cas non-lipsehitzien. In Seminaire de Probabilites XII, vol­urne 649 of Lecture Notes in Mathematics, pages 114-131. Springer-Verlag, Berlin, 1978.

256. J. A. Yan. Caraeterisation d'une classe d'ensembles eonvexes de L 1 ou H 1 • In Seminaire de Probabilites XIV, volume 784 of Lecture Notes in Mathematics, pages 220-222. Springer-Verlag, Berlin, 1980.

257. Ch. Yoeurp. Deeomposition des martingales loeales et formules exponentielles. In Seminaire de Probabilites X, volume 511 of Lecture Notes in Mathematics, pages 432-480. Springer-Verlag, Berlin, 1976.

258. Ch. Yoeurp. Solution explicite de l'equation Zt = 1+ J~ IZs-ldXs. In Seminaire de Probabilites XIII, volume 721 of Lecture Notes in Mathematics, pages 614-619. Springer-Verlag, Berlin, 1979.

259. Ch. Yoeurp. Sur la derivation des integrales stoehastiques. In Seminaire de Probabilites XIV, volume 784 of Lecture Notes in Mathematics, pages 249-253. Springer-Verlag, Berlin, 1980.

260. Ch. Yoeurp and M. Yor. Espaee orthogonal a une semimartingale: applieations. Unpublished, 1977.

261. M. Yor. Sur les integrales stoehastiques optionnelles et une suite remarquable de formules exponentielles. In Seminaire de Probabilites X, volume 511 of Lecture Notes in Mathematics, pages 481-500. Springer-Verlag, Berlin, 1976.

262. M. Yor. Apropos d'un lemme de Ch. Yoeurp. In Seminaire de Probabilites XI, volume 581 of Lecture Notes in Mathematics, pages 493-501. Springer-Verlag, Berlin, 1977.

410 References

263. M. Yor. Sur quelques approximations d'integrales stochastiques. In Seminaire de Probabilites XI, volume 581 of Leeture Notes in Mathematies, pages 518-528. Springer-Verlag, Berlin, 1977.

264. M. Yor. Grossissement d'une filtration et semi-martingales: theoremes gener­aux. In 8eminaire de Probabilites XII, volume 649 of Leeture Notes in Mathe­maties, pages 61-69. Springer-Verlag, Berlin, 1978.

265. M. Yor. Inegalites entre processus minces et applications. C. R. Aead. Sei. Paris Sero A-B, 286:A799-A801, 1978.

266. M. Yor. Rappels et preliminaires generaux. Asterisque, 52-53:17-22,1978. 267. M. Yor. Sur la continuite des temps locaux assodes a certaines semi­

martingales. In Temps Locaux, volume 52 and 53 of Asterisque, pages 23-36. Sodete Mathematique de France, Paris, 1978.

268. M. Yor. Un exemple de processus qui n'est pas une semi-martingale. In Temps Loeaux, volume 52 and 53 of Asterisque, pages 219-222. Sodete Mathematique de France, Paris, 1978.

269. M. Yor. Application d'un lemme de Jeulin au grossissement de la filtration brownienne. In Seminaire de Probabilites XIV, volume 784 of Leeture Notes in Mathematics, pages 189-199. Springer-Verlag, Berlin, 1980.

270. M. Yor. Remarques sur une formule de Paul Levy. In Seminaire de Probabilites XIV, volume 784 of Leeture Notes in Mathematies, pages 343-346. Springer­Verlag, Berlin, 1980.

271. M. Yor. Some Aspeets 01 Brownian Motion. Part II: Some Reeent Martingale Problems. Birkhäuser, Basel, Switzerland, 1997.

272. W. Zheng. Une remarque sur une meme integrale stochastique calculee dans deux filtrations. In Seminaire de Probabilites XVIII, volume 1059 of Leeture Notes in Mathematies, pages 172-178. Springer-Verlag, Berlin, 1984.

Subject Index

a-sliceable 254 EMO see bounded mean oscillation ]Ei' optional projection 377 ]Ei' special, <G semimartingale 377 F-S see Fisk-Stratonovich FV see finite variation process H 2 norm 156 HP norm 195 HP

(pre)locally in 253 converge (pre)locally in 266 norm 251

/iP (pre)locally in 253 converge (pre)locally in 266 norm 250

ucp see uniform convergence on compacts in probability

absolutely continuous 133 Absolutely Continuous Compensators

Theorem 194 absolutely continuous space 193 accessible stopping time 104 adapted process

decomposable 55 definition of 3 space lL of with caglad paths 157 with bounded caglad paths 56 with cadlag paths 56 with caglad paths 56

angle bracket 125 announcing sequence for stopping time

104

approximation of the compensator by Laplacians 152

arcsine law 233 associated jump process 27 asymptotic martingale (AMART) 222 autonomous 256,299 Azema's martingale 208

and last exit from zero 233 definition of 233 local time of 235

Backwards Convergence Theorem 9 Banach-Steinhaus Theorem 43 Bessel process 361 Bichteler-Dellacherie Theorem 146 Bougerol's identity 290 Bouleau-Yor formula 230,232 bounded functional Lipschitz 263 bounded jumps 25 bounded mean oscillation

EMO norm 197 EMO space of martingales 197 Duality Theorem 201

bracket process 66 Brownian bridge 97,306,392 Brownian motion 17

]Ei' Brownian motion 17 absolute value of 220 arcsine law for last exit from zero

233 Azema's martingale 233 Brownian bridge see Brownian

bridge

412 Subject Index

completed natural filtration right continuous 19

conditional quadratic variation of 125

continuous modification 17 continuous paths 225 definition of 17 Fisk-Stratonovieh exponential of

287 Fisk-Stratonovieh integral for 294 geometrie 86 Girsanov-Meyer Theorem and 144 Itö integral for 78 Levy's characterization of 88 Levy's Theorem (martingale

characterization) 86,87 last exit from zero 233 last exit time 46 local time of 220, 230 martingale characterization (Levy's

Theorem) 86,87 martingale representation for 189 martingales as time change of 88,

189 maximum process 23 on the sphere 288 Ornstein-Uhlenbeck 98,305 quadratie variation of 17,67, 125 reflection principle 24, 233 reversibility for integrals 389 semimartingale 55 skew Brownian motion 247 standard 17 starting at x 17 stochastie area formula 89 stochastie exponential for see

geometrie Brownian motion stochastie integral exists 175 strong Markov property for 23 Tanaka's formula 220 tied down or pinned see Brownian

bridge time change of 88 unbounded variation 19 white noise 143, 249

Burkholder's inequality 226 Burkholder-Davis-Gundy inequalities

195

cadlag 4 caglad 4 canonieal decomposition 131,156 censored data 123 centering function 32 change of time 88, 192

exercises in Chap. II 99 in stochastic integrals (Le Jan's

formula) 247 Lebesgue's formula 192

change of variables see Itö's formula, see Meyer-Itö formula

Change of Variables Theorem for continuous FV processes 41 for right continuous FV processes

78 classieal semimartingale 102,129,146 Class D 107 closed martingale 8 Comparison Theorem 331 compensated Poisson process 31,42,

65 compensator 120

Absolutely Continuous Compensators Theorem 194

Knight's compensator calculation method 153

compensator of l{t2:L} for lF 379 compound Poisson process 33 conditional quadratie variation 70,

124 Brownian motion 125 polarization identity 125

continuous local martingale part of semimartingale 70,226

continuous martingale part 193 converge (pre)locally in HP, fiP 266 convex functions --

and Meyer-Itö formula 218 of semimartingale 214

countably-valued random variables corollary 373

counting process 12 explosion time 13 without explosions 13

crude hazard rate 124

decomposable adapted process 55,101 diffeomorphism of IRn 325

diffusion 249,297,304 examples 305 rotation invariant 288

diffusion coefficient 304 Dirichlet process 222 discrete Laplacian approximations 153 Doleans-Dade exponential see

stochastic exponential Dominated Convergence Theorem for

stochastic integrals in H T 273 in ucp 176

Doob dass see Class D Doob decomposition 106 Doob's maximal quadratic inequality

11 Doob's Optional Sampling Theorem 9 Doob--Meyer Decomposition Theorem

case without Class D 116 general case 112 totally inaccessible jumps 107

drift coefficient 145,304 removal of 138

Duality Theorem 201 Dynkin's expectation formula 356 Dynkin's formula 56,356

Einstein convention 312 Emery's example of stochastic integral

behaving badly 178 Emery's inequality 252 Emery's structure equation see

structure equation Emery-Perkins Theorem 245 enveloping sequence for stopping time

104 equivalent probability measures 133 Euler method of approximation 359 evanescent 160 example

Emery's example of stochastic integral behaving badly 178

expansion via end of SDE 375 Gaussian expansions 374 hazard rates and censored data 123 Itö's example 374 reversibility for Brownian integrals

389

Subject Index 413

reversibility for Levy process integrals 388

example of local martingale that is not martingale 37,74

example of process that is not semimartingale 221

example of stochastic integral that is not local martingale 178

examples of diffusions 305 Existence of Solutions of Structure

Equation Theorem 204 expanded filtration 378 expansion by a natural exponential r. v.

394 expansion via end of SDE 375 explosion time 13,260,310 extended Gronwall's inequality 358 extremal point 185

Fefferman's inequality 197 strengthened 197

Feller process 35 filtration

countably-valued random variables corollary 373

definition of 3 expansion by a natural exponential

r.v. 394 filtration shrinkage 375 Filtration Shrinkage Theorem 377 Gaussian expansions 374 independence corollary 373 Itö's example 374 natural 16 progressive expansion 364,378 quasi left continuous 150,191 right continuous 3

filtration shrinkage 375 Filtration Shrinkage Theorem 377 finite quadratic variation 277 finite variation process

definition of 39, 101 integrable variation 112

Fisk-Stratonovich acceptable 284 approximation as limit of sums integral 82, 277

291

integral for Brownian motion Integration by Parts Theorem

294 284

414 Subject Index

Itö's circle 82 Itö's formula 283 stochastie exponential 286

flow of SDE 307 of solution of SDE 390 strongly injective 317 weakly injective 317

Fubini's Theorem for stochastie integrals 210, 211

function space 308 functional Lipschitz 256 fundamental L martingale 380 fundamental sequence 37 Fundamental Theorem of Local

Martingales 126

Gamma process 33 Gaussian expansions 374 generalized Itö's formula 278 generator of stable subspace 181 geometrie Brownian motion 86 geometric inversion mapping 324 Girsanov-Meyer Theorem 134

predietable version 135 graph of stopping time 104 Gronwall's inequality 349,358

extended 358

Hadamard's Theorem 337 Hahn-Banach Theorem 201 hazard rates 123 hitting time 4 Hölder continuous 100 honest random variable 381 Hunt process 36 Hypothesis H 396 Hypothesis Hf 396 Hypothesis A 225

increasing process 39 independence corollary 373 independent increments

of Uivy process 20 of Poisson process 13

index of stable law 34 indistinguishable 3, 60 infinitesimal generator 355

injective flow see strongly injective flow

integrable process 379 integrable variation process 112 Integration by Parts Theorem

for Fisk-Stratonovieh integrals 284 for semimartingales 68, 83

intrinsie Levy process 20 Itö integral see stochastie integral Itö's circle 82 Itö's formula

and Cl functions 221 for an n-tuple of semimartingales 81 for complex semimartingales 83, 84 for continuous semimartingales 81 for Fisk-Stratonovich integrals 283 for semimartingales 78 generalized 278

Itö-Meyer formula see Meyer-Itö formula

Itö's example 374 Itö's Theorem extended to Levy

processes 364

Jacod's Countable Expansion 374 Jacod's Countable Expansion Theorem

53,364 Jacod's criterion 371 Jacod-Yor Theorem on martingale

representation 201 Jensen's Inequality 11 Jeulin's Lemma 368

Kazamaki's criterion 141 Knight's compensator calculation

method 153 Kolmogorov's continuity criterion 225 Kolmogorov's Lemma 223 Kronecker's delta 312 Kunita-Watanabe inequality 69, 150

Levy Decomposition Theorem 31 Levy measure 26 Levy process

associated jump process 27 bounded jumps 25 cadlag version 25 centering function 32 definition of 20

has finite moments of all orders 25 intrinsic 20 is a semimartingale 55 Itö's Theorem for 364 Levy measure 26 refiection principle 49 reversibility for integrals 388 strong Markov property for 23 symmetrie 49

Levy's arcsine law 233 Levy's stochastic area formula 89 Levy's stochastic area process 89 Levy's Theorem characterizing

Brownian motion 86,87 Levy-Khintchine formula 32 last exit from zero 233 Le Gall 's Theorem 246 Le Jan's Theorem 126 Lebesgue's change of time formula 192 Lenglart's Inclusion Theorem 179 Lenglart-Girsanov Theorem 137 linkage operators 336 Lipschitz

bounded functional 263 definition of 256, 299 functional 256 locally 257,260,309 process 256,317 random 256

local behavior of stochastic integral 62,167,172

local behavior of the stochastic integral at random times 383

local convergence in HP, fiP 266 local martingale --

BMO 197 cadlag martingale is 37 compensator of 120 condition to be a martingale 38, 73 continuous part of semimartingale

226 decomposable 128 definition of 37 example that is not a martingale

37,74 fundamental sequence for 37 Fundamental Theorem 126 intervals of constancy 71, 75 not locally square integrable 128

Subject Index 415

not preserved under shrinkage of filtration 128

pre-stopped 174 preserved by stochastic integration

see stochastic integral, preserves reducing stopping time for 37 time change of Brownian motion 88

local property 38,164 local time

and Bouleau-Yor formula 232 and change of variables formula

see Meyer-Itö formula, see Meyer-Tanaka formula

and delta functions 220 continuous in t 216 definition of L~ 216 discontinuous, example of 229 method of 246 occupation time density 219, 230 of Azema's martingale 235 of Brownian motion 220 of semimartingale 216 regularity in space 228 support of 217

locally bounded process 166 locally in HP, fiP 253 locally integrahle process 367 locally integrable variation process

112 locally Lipschitz 257,260,309 locally special semimartingale 151

Memin's criterion for exponential martingales 358

Metivier-Pellaumail inequality 358 Metivier-Pellaumail method 358 Markov process

definition of 34 diffusion 249,297,304 Dynkin's expectation formula 356 Dynkin's formula 56 equivalence of definitions 35 infinitesimal generator 355 simple 298 strong 299 time homogeneous transition function transition semigroup

martingale

35,298 35,298

298

416 Subjeet Index

L2 projeetion 9 ']-fP norm 195 EMO 197 asymptotic (AMART) 222 Azema's 208, 233 Baekwards Convergenee Theorem 9 Burkholder's inequality 226 eadlag modifieation 8 closed 8 eontinuous martingale part 193 definition of 7 Doob's inequalities 11 fundamental L martingale 380 in L2 180 Jaeod-Yor Theorem 201 loeal 37 Martingale Convergenee Theorem 8 martingale representation 206 maximal quadratic inequality 11 measures M 2 (A) 184 not loeally square integrable 128 Optional Sampling Theorem 9 orthogonal 181 predictable representation property

184 purely diseontinuous part 193 quasimartingale 118 sigma martingale 237 space M 2 of 180 square integrable 73, 74 stronglyorthogonal 181 submartingale 7 supermartingale 7 uniformly integrable 8 with exaetly one jump 126

Martingale Convergenee Theorem 8 martingale representation 206 mathematical finanee theory 138 maximal quadratie inequality 11 measurable O"-algebra 103 Meyer's Theorem 105 Meyer-Ito formula 218

and Cl functions 221 extant seeond derivative 221

Meyer-Tanaka formula 220 modifieation 3 Monotone Class Theorem 7 monotone veetor spaee 7

closed under uniform eonvergenee 7

multiplieative eolleetion of functions 7

natural filtration 16 natural proeess 113 net hazard rate 124 Novikov's eriterion 141

oblique bracket 125 oeeupation time density 219,230 optional O"-algebra 102,380 optional projeetion 375, 377 Optional Sampling Theorem 9 Ornstein-Uhlenbeek proeess 98,305 orthogonal martingales 181 Ouknine's formula 245

partition 118 path spaee 144 path-by-path eontinuous part 70 Perkins-Emery Theorem 245 Picard iteration 261 pinned Brownian motion 306 Pitman's Theorem 245 Poisson proeess

arrival rate 15 eompensated 31,42,65, 120 eompound 33 definition of 13 independent inerements 13 intensity 15 stationary inerements 13

polarization identity 66, 125, 277 potential 106, 152 pre-stopped loeal martingale 174 predictable O"-algebra 102,156 predictable eompensator of l{t~L} for lF

379 predictable proeess

predictable O"-algebra 102 simple 51

predictable projeetion 376 predietable representation property

184 predictable stopping time 104 predictably measurable proeess 102,

107 preloeal eonvergenee in HP, s..p 266 preloeally in HP, gP 253 -

preserved by stochastic integration see stochastic integral, preserves

process Lipschitz 256, 317 process stopped at T 10, 37 progressive a-algebra 103 progressive expansion 364, 378 progressive process 103 projections

optional projection 375 predictable projection 376

property holds locally 38, 164, 253 property holds prelocally 164,253 purely discontinuous part of martingale

193

quadratic covariation 66, 386 polarization identity 66,277

quadratic covariation process 277 quadratic pure jump 71 quadratic variation

conditional 124 continuous part of 70, 277 covariation 66 finite 277 of a semimartingale 66 polarization identity 66,277

quasi left continuous filtration 150, 191

quasimartingale 118

random Lipschitz 256 random partition

definition of 64 tending to the identity 64

Rao's Theorem 119 Ray-Knight Theorem 230 reduced by stopping time 37 reflection principle

for Brownian motion 24, 233 for Levy processes 49 for stochastic area process 92

regular conditional distribution 371 regular supermartingale 152 representation property 184 reversibility for Brownian integrals

389 reversibility for Levy process integrals

388 reversible semimartingale 386

Subject Index 417

Riemann-Stieltjes integral 41 right continuous filtration 3 right stochastic exponential 325, 326 right stochastic integral 325 rotation invariant diffusion 288

sampie paths of stochastic process 4 scaling property 243 self-sufficient filtration 396 semimartingale

(lF, G) reversible 386 (re, X) integrable 165 1t2 norm 156 absolute value of see Meyer-Tanaka

formula bracket process 66 canonical decomposition 131 classical 102, 129, 146 continuous local martingale part 70,

226 convex function of 214 definition of 52 equivalent norm 250 example of process that is not 221 Integration by Parts Theorem 68,

83 local martingale as 129 local time of 216 locally special 151 preserved by stochastic integration

see stochastic integral, preserves quadratic pure jump 71 quadratic variation of 66 quasimartingale as 129 space 1t2 of 156 special 130, 156 stochastic exponential of 85 submartingale as 129 supermartingale as 129 topology 270 total 52

semimartingale topology 270 separable measurable space 191 sharp bracket 125 sigma martingale 237

preserved by stochastic integration see stochastic integral, preserves

sign function 216 signal detection 142

418 Subjeet Index

simple Markov proeess 298 simple predietable proeess 51 Skew Brownian motion 247 Skorohod topology 225 Skorohod's Lemma 244 smallest filtration making r.v. a

stopping time 121,378 special semimartingale 130,156

eanonieal deeomposition 131 square integrable martingale 180

preserved by stoehastic integration see stoehastie integral, preserves

stable law 34 index 34

stable proeess defintion of 34 sealing properties 34 symmetrie 34

stable subspaee generated 181 of M 2 180 of re 202

stable under stopping 180 standard Borel spaee 370 stationary Gaussian proeess 305 stationary inerements

of Levy proeess 20 of Poisson proeess 13

statistical eommunication theory 142 Stieltjes integral see Riemann-Stieltjes

integral stoehastic area formula 89 stoehastic area proeess

definition of 89 density 91 Levy's formula 89 properties 92 refieetion principle 92

stoehastie differential equation 261, 328

explicit solutions 290 fiow of 307 fiow of solution 390 weak solution 204, 246 weak uniqueness 204

stoehastic exponential appproximation of 276 as a diffusion 305 definition of 85

Fisk-Stratonovieh 286 for Brownian motion see geometrie

Brownian motion for semimartingale 85 invertibility of 342 right 325, 326 uniqueness of 261

stoehastic integral X integrable, L(X) 165 Assoeiativity Theorem 62,161,167 behavior of jumps 60, 160, 167 does not preserve FV proeesses in

general 246 Dominated Convergenee Theorem in

H r 273 Dominated Convergenee Theorem in

uep 176 example that is not loeal martingale

178 for lL 59 for bP 158 for S 58 for P 165 for Brownian motion 78 Fubini's Theorem 210,211 loeal behavior at random times 383 loeal behavior of 62, 167, 172 preserves

FV proeesses, integrands in lL 63 eontinuous loeal martingales 175 loeal martingales 129, 173, 176 loeally square integrable loeal

martingales 63, 173 semimartingales 63 sigma martingales 238 square integrable martingales 161,

173 right 325 said to exist 165

stoehastie proeess adapted 3 deeomposable 55, 101 definition of 3 finite variation 39,101 inereasing 39 indistinguishable loeally bounded loeally integrable modifieation of 3

3,60 166

367

natural 113 potential 106 sample paths of 4 stopped 10,37 strong Markov 36

stopped process 10,37 stopping time

a-algebra 5, 105 accessible 104 announcing sequence 104 change of time 99 definition of 3 enveloping sequence 104 explosion time 13, 260, 310 fundamental sequence of 37 graph 104 hitting time 4 hitting time of Borel set is 5 predietable 104 reducing 37 totally inaccessible 104

Stratonovieh integral see Fisk­Stratonovieh, integral

Stratonovich Integration by Parts Theorem 284

strengthened Fefferman inequality 197 Strieker's Theorem 53 strong Markov process 36, 299 strong Markov property 36,299

for Brownian motion 23 for Levy process 23

strongly injective flow 317 strongly orthogonal martingales 181 structure equation 204, 243

Existence of Solutions Theorem 204 submartingale 7

Subject Index 419

supermartingale 7 closed 9 regular 152

symmetrie Levy process 49 symmetrie stable process 34

Tanaka's formula 220 tied down Brownian motion 306 time change 192 time change of Brownian motion 88 time homogeneous Markov process

298 time reversal 386 total semimartingale 52 total variation process 40 totally inaccessible stopping time 104 transition function for Markov process

298 transition semigroup 298 truncation operators 271

uniform convergence on compacts in probability 57

uniformly integrable martingale 8 usual hypotheses 3,298

variation 118 variation along T 118

weak solution of SDE 204, 246 weak uniqueness of SDE 204 weakly injective flow 317 white noise 142,249 Wiener measure 144 Wiener process 139,142

Yoeurp's lemma 114

Stochastic Modelling and Applied Probability formerly entitled Applications of Mathematics

43 Yong/Zhon, Stochastic Controls. HamiItonian Systems and HJB Equations (1999) 44 Serfozo, Introduction to Stochastic Networks (1999) 45 Steele, Invitation to Stochastic Calculus (2000) 46 Chen/Yao, Fundamentals of Queuing Networks: Performance, Asymptotics, and

Optimization (2001) 47 Kushner, Heavy Trafiic Analysis of Controlled Queuing and Communications

Networks (2001) 48 Fernholz, Stochastic Portfolio Theory (2002) 49 Kabanov/Pergamenshchikov, Two-Scale Stochastic Systems (2003) 50 Han, Information-Spectrum Methods in Information Theory (2003) 51 Asmussen, Applied ProbabiIity and Queues (2nd ed. 2003) 52 Robert, Stochastic Networks and Queues (2003) 53 Glasserman, Monte Carlo Methods in Financial Engineering (2004) 54 Sethi/Zhangl Zhang, Average-Cost Control of Stochastic Manufacturing Systems

(2004) 55 Yin/Zhang, Discrete-Time Markov Chains: Two-Time-Scale Methods and

Applications (2004)