Bibliography

Alloin, C. [1] "Processus previsibles optimaux associes a un processus stochas• tique." Cahiers cent re Etud. Rech. Oper. 11 (1969) 92-103. Ambrose,W. [1] "On measurable stochastic processes." Trans. Am. Math. Soc. 47(1940) 66-79. Andersen, E.S., and Jessen, B. [1] "Some limit theorems on in an abstract set."Danske Vid. Selsk. Mat.-Fys. Medd. 22(14)(1946) 29pp. [2] "On the introduction of measures in infinite product sets." ibid. 25(4)(1948) 8pp. Anderson, R.F. [1] "Diffusions with second order boundary conditions, Parts I - I1." Indiana Univ. Math. J. 25(1976) 367-395; 403-44l. Auslander, L., and MacKenzie, R. E. [1] Introduction to Differential Manifolds. McGraw-Hill, New York, 1963. Austin, D. G. [1] "A sample function property ofmartingales." Ann. Math. Statist. 37(1966) 1396-1397. Bartle, R. G. [1] "A bilinear vector ." Studia Math. 15(1956) 337-352. Bell, D. R. [1] The Malliavin . Pitman Math. Mono. 34 London, 1987. Belopol'skaya, Va. 1., and Dalecky, Yu. L. [1] Stochastic Equations and Differential Geometry. Kluwer Acad. Publ. Boston, MA 1990. Bichteler, K. [1] "Stochastic integration and LP-theory of ." Ann. Prob.9(1981) 49-89. Bismut, J.-M. [1] "Martingales, the , and hypoellipticity under general Hörmander conditions." Z. Wahrs. 56(1981) 469-505. [2] Large Deviations and the Malliavin Calculus. Birkhauser, 1984.

589 590

Blake, L. H. [1] "A generalization ofmartingales and two eonsequent eonvergenee theorems." Paeifie J. Math. 35 (1970) 279-283. Blumenthal, R. M., and Getoor, R. K. [1] Markov Proeesses and Potential Theory. Aeademie press. New York, 1968. Boehner, S. [1] "Stoehastie proeesses." Ann. Math. 48(2)(1947) 1014-106l. [2] Harmonie Analysis and the Theory of Probability. Univ. of Calif. Press, Berkeley, CA, 1955. [3] "Partial ordering in the theory of martingales." Ann. Math. 62(2) (1955) 162-169. [4] "Stationarity, boundedness, almost periodieity of random valued functions." Proe. Third Berkley Symp. Math. Statist. Prob. 2(1956) 7-27. Bonami, A., Karoui, N., Roynette, B., and Reinhard, H. [1] "Proeessus de diffusion assoeie a un operateur elliptique degenere." Ann. Inst. H. Poineare, 7(1971) 31-80. Borehers, D. R. [1] Seeond Order Stoehastie Differential Equations and Related Ito proeesses. Ph.D. thesis, Carnegie-Mellon Univ. Pittsburgh, 1964. Bourbaki, N. [1] Elenents de Mathematique VI. Chapitre IX (also Chaps. 3-5). Hermann, Paris. 1969. Brennan, M. D. [1] "Planar semimartingales." J. Multivar. Anal. 9(1979) 465-486. [2] "Riemann-Stieltjes quasimartingale integration." J. Multivar. Anal. 10(1980) 517-538. Brooks, J. K., and Dineuleanu, N. [1] "Stoehastie integration in Banaeh spaees." Seminar on Stoehastie Proeesses, Birkhauser, Basel, (1991) 27-115. Burkholder, D. 1. [1] "Distribution function inequalities for martingales." Ann. Prob. 1(1973) 19-42. Burkholder, D. 1., Davis, B. J., and Gundy, R. F. [1] "Integral inequalities for eonvex functions of operators on martin• gales." Proe. Sixth Berkeley Symp. Math. Statist. Prob., Univ. of Bibliography 591

Calif. Press, 2(1972) 223-240. Cabaiia, E. M. [1] "Stochastic integration in separable Hilbert spaces." Montevideo Publ. Inst. Mat. Estad. 4(1966) 1-27. Cairoli, R. [1] "Une inequalite pour martingales a indices multiples et ses appli• cations." Lect. Notes Math. 124(1970) 49-80. [2] "Decomposition de processus a indices double." ibid. 191(1971) 37-57. Cairoli, R., and Walsh, J. B. [1) "Stochastic integrals in the plane." Acta Math. 134(1975) 111- 183. Carmona, R.A., and Nualart, D. [1) Nonlinear Stochastic Integrators, Equations and Flows. Gordon and Breach, New York, 1990. Cartan, H. [1] Elementary Theory of Analytic Functions of One or Several Com• plex Variables. Addison-Wesley, Reading, MA 1963. Cartier, P. [1] "Introduction a l'etude des movements browniens a plusieurs parameters." Lect. Notes Math. 191(1971) 58-75. Chac6n, R. V. [1] "A'stopped' proof of convergence." Adv. Math. 14(1974) 365-368. Chang, D. K., and Rao, M. M. [1] "Bimeasures and nonstationary processes."in Real and Stochastic Analysis, Wiley, New York, (1986) 7-118. Choksi, J. R. [1) "Inverse limits ofmeasure spaces." Proc. London Math. Soc. 8(3) (1958) 321-342. Choquet, G. [1] "Ensembles K-analytiques et K-sousliniens, cas general et cas metrique." (et autrer articles) Ann. Inst. Fourier Grenoble, 9(1959) 75-109. Chow, Y. S.

[1) "Martingales In a o--n.nite measure space indexed by directed sets." Trans. Am. Math. Soc. 97(1960) 254-285. 592

[2] "A martingale inequality and the ." Proc. Am. Math. Soc. 11(1960) 107-11l. [3] "Convergence of sums of squares of martingale differences." Ann. Math. Statist. 39(1968) 123-133. [4] "Convergence theorems for martingales." Z. Wahrs. 1(1963) 340- 346. Chung, K. L., and Doob, J. L. [1] "Fields, optimality and measurability." Am. J. Math. 87(1965) 397-424. Clarkson, J. A., and Adams, C. R. [1] "On definitions of bounded variation for functions of two vari• ables." Trans. Am. Math. Soc. 35(1933) 824-854. Coddington, E.A. and Levinson, N. [1] Theory of Ordinary Differential Equations. McGraw-Hill, New York, 1955. Cornea, A., and Licea, G. [1] "General optional sampling of super martingales." Rev. Roum. Math. Pures Appl. 10(1965) 1379-1367. Courrege, P., and Prioret, P. [1] "Temps d'arret d'unefonction aleatoire." Publ. Inst. Statist. Univ. Paris, 14(1965) 245-274. Cramer, H. [1] Mathematical Methods of . Princeton Univ. Press, Prince• ton, NJ, 1946. Cuculescu, 1. [1] "Spectral families and stochastic integrals." Rev. Roum. Math. Pures appl. 15(1970) 201-22l. Dalecky, Yu. L. [1] "Infinite dimensional elliptic operators and parabolic equations connected with them." Russian Math. Surveys, 22(4)(1967) 1- 53. Dambis, K. E. [1] "On the decomposition of continuous submartingales." Th. Prob. Appl. 10(1965) 401-410. Darling, R.W.R. [1] Constructing Nonlinear Stochastic Flows. Memoirs Am. Math. Soc., 376, (1987), 1-97. Bibliography 593

Davis, B. J. [1] "On the integrability of the martingale square function." Israel J. Math. 8(1970) 187-190. Davis, M. [1] Applied Nonstand Analysis. Wiley-Interscience, New York, 1977. de la Vallee Poussin, C.J. [1] "Sur l'integrale de Lebesgue." Trans. Am. Math. Soc. 16 (1915) 435-50l. Dellacherie, C. [1] Capacites et Processus Stochastiques. Springer-Verlag, Berlin, 1972. [2] "Un survol de la theorie de l'integrale stochastique." Lect. Notes Math. 794(1980) 368-395. Dinculeanu, N. [1] Vector Measures. Pergamon Press, London, 1967. [2] "Conditional expectations for general measure spaces." J. Multi• var. Anal. 1(1971) 347-364. [3] "Vector valued stochastic processes, I-V." I,J. Th. Prob. 1(1988) 149-169; II, Proc. Am. Math. Soc. 102393-401; III, Sem. Stoch. Processes, Birkhauser (1988) 93-132; IV, J. Math. Anal. 142(1989) 144-161; V, Proc. Am. Math. Soc. 104(1988) 625-63l. [4] "Linear operators on spaces oftotally measurable functions." Rev. Roum. Math. Pures Appl. 10(1965) 1493-1524. Dinculeanu, N., and Rao, M. M. [1] "Contractive projections and conditional expectations." J. Mul• tivar. Anal. 2(1972) 362-38l. Doleans-Dade, C. (=Doleans, C.) [1] "Variation quadratique des martingales continue a droit." Ann. Math. Statist. 40(1969) 284-289. [2] "Quelques applications de la formule de changement de variables pour les semimartingales." Z. Wahrs. 16(1970) 181-194. [3] "Existence du processus croissant natural associe a un potentiel de la dass (D)." Z. Wahrs. 9(1968) 309-314. [4] "On the existence and unicity of solutions of stochastic integral equations." Z. Wahrs. 36(1976) 93-10l. Doleans-Dade, C., and Meyer, P. A. 594

[1] "Integrales stochastique par rapport aux martingales locales." Lect. Notes Math. 124(1970) 77-104. Doob, J. L. [1] Stochastic Processes. Wiley, New York, 1953. [2] "Notes on martingale theory." Proc. Fourth Berkley Symp. Math. Statist. Prob. 2( 1961) 95-102. [3] Classical Potential Theory and its Probabilistic Counterpart. Springer-Verlag, Berlin, 1984. Dozzi, M. [1] Stochastic Processes with a Multidimensional Parameter. Res.N otes Math. Pitman, London, 1989. Dubins, L. E. [1] "Conditional prob ability distributions in the wide sense." Proc. Am. Math. Soc. 8(1957) 1088-1092. Dubins,1. E., and Schwarz, G. [1] "On continuous martingales." Proc. Nat'l. Acad. Sei. 53(1965) 913-916. Dunford, N., and Schwartz, J. T. [1] Linear Operators, Part I: General Theory. Wiley-Interscience, New York, 1958. Dym, H. [1] "Stationary measures for the flow of a linear differential equation driven by ." Trans. Am. Math. Soc. 123(1966) 130- 164. Dynkin, E. B. [1] Foundations of the Theory of Markov Process. Pergamon Press, London, 1960. [2] Markov Process (2 Vols.), Springer-Verlag, New York, 1965. Eberlein, W. F. [1] "Abstract ergodie theorems and weak almost periodic functions." Trans. Am. Math. Soc. 67(1949) 217-240. Edgar, G. A., and Sucheston, L. [1] "Amarts: a class of asymptotic martingales." J. Multivar. Anal. 6(1976) 193-221, 572-591. [2] Stopping Times and Directed Processes. Cambridge Univ. Press, London, 1993. Emery, M. Bibliography 595

[1) in Manifolds. Springer-Verlag, New York, 1989. Fefferman, C. L. [1) "Characterization ofbounded mean oscillation." BuU. Am. Math. Soc. 77(1971) 587-588. Fefferman, C. L., and Stein, E. M. [1) "}iP-spaces of several variables." Acta. Math. 129(1972) 137-193. Feldman, J. [1) "Equivalence and perpendicularity of Gaussian processes. " Pacific J. Math. 8(1959) 699-708. FeUer, W. [1) Theory of Probability and its Applications. Vo1.2, Wiley, New York, 1966. [2) "Non-Markovian processes with the semigroup property." Ann. Math. Statist. 30(1959) 1252-1253. [3) "The parabolic partial differential equations and the associated semigroup oftransformations." Ann. Math. 55(2)(1952) 468-519. Fillmore, P. A. [1) "On topology induced by measure." Proc. Am. Math. Soc. 17 854-857. Finlayson, H. C. [1) "Measurability of norm proved by Haar functions." Proc. Am. Math. Soc. 53( 1975) 334-336. [2) "Two classical examples of Gross' abstract Wiener measure." Proc. Am. Math. Soc. 53(1975) 337-340. [3) "Gross' abstract Wiener measure on G[O, 00]." Proc. Am. Math. Soc. 57(1976) 297-298. Fisk, D. L. [1) "Quasi-martingales." Trans. Am. Math. Soc. 120(1965) 369-389. [2) "Sample of sample continuous second order martingales." Z. Wahrs. 6(1966) 273-278. Flanders, H. [1) Differential Forms with Applications to Physical Sciences. Dover, New York, 1989. Föllmer, H. [1) "On the representation of semimartingales." Ann. Prob. 1(1973) 580-589. [2) "Stochastic holomorphy." Math. Ann. 207(1974) 245-255. 596

[3] "Quasimartingales a deux indices." C. R. Acad. Sci., Paris, Sero A, 288(1979) 61-64. Freidlin, M. [1] Functional Integration and Partial Differential Equations. Prince• ton Univ. Press, Princeton, NJ, 1985. Gangolli, R. [1] Positive definite kerneis on homogeneous spaces and certain sto• chastic processes related to Levy's Brownian motion." Ann. Inst. H. Poincre, 3(1967) 121-225. Garsia, A. M. [1] "The Burgess Davis inequalities via Fefferman's inequality." Ark. Math. 11(1973) 229-237. [2] Martingale Inequalities. Benjaman, Inc., Reading, MA, 1973. [3] "A convex function inequality for martingales." Ann. Prob. 1(1973) 171-174. Gel'fand,1. M., and Vilenkin, N. Va. [1] Generalized Functions, Vol. 4, Acad. Press, New York, 1964. Getoor, R. M., and Sharpe, M. J. [1] "Conformal martingales." Invent. Math. 16(1972) 271-308. Gikhman,1. 1., and Skorokhod, A.V. [1] Introduction to the Theory of Random Processes. Saunders, Philadelphia, PA, 1969. [2] "On the densities of probability measures in function spaces." Russian Math. Surveys, 21(6) 83-156. Girsanov, 1. V. [1] "On transforming a certain class of stochastic processes by ab• solutely continuous substitution of measures." Th. Prob. Appl. 5 285-30l. Girshick, M. A., and Savage, L. J. [1] "Bayes and minimax estimates for quadratic loss." Proc. Second Berkely Symp. Math. Statist. Prob. (1951) 285-30l. Goldberg, S. 1. [1] Curvature and Homology. Dover, New York, 1982. Goldstein, J. A. [1] "Abstract evolution equations." Trans. Am. Math. Soc. 141(1969) 159-185. [2] "Second order Itö processes." Nagoya Math. J. 36(1969) 27-63. Bibliography 597

[3] "An existence theorem for linear stoehastie differential equations." J. Diff. Eq. 3(1967) 78-87. Gould, G. G. [1] "Integration over vector-valued measures." Proe. London Math. Soe. 15(3)(1965) 193-225. Green, M. L. [1] Multiparameter Integrals and Boundedness Prin• ciples. Ph. D. thesis, Univ. Calif., Riverside, 1995. Gross, L. [1] "Measurable functions on a Hilbert spaee." Trans. Am. Math. Soe. 105(1962) 372-390. [2] "Abstract Wiener spaees." Proe. Fifth Berkeley Symp. Math. Statist. Prob. 2(1967) 31-42. [3] "Potential theory on Hilbert spaee." J. Funet. Anal. 1(1967) 123- 18l. Guggenheimer, H. W. [1] Differential Geometry. Dover, New York, 1977. Gundy, R. F. [1] "A deeomposition for L1-bounded martingales." Ann. Math. Statist. 39( 1968) 134-138. Hajek, J. [1] "On a property of of any stoehastie proeess." Ceeh. Math, J, 8(2)(1958) 610-618. Hajek, J., and Renyi, A. [1] "Generalization of an inequality of Kolmogorov." Aeta Math. Aead. Sei., Hung. 6(1955) 281-283. Halmos, P. R. [1] Measure Theory. Van Nostrand. Prineeton, NJ, 1950. Hardy, G. H., Littlewood, J. E., and P6lya, G. [1] Inequalities. Cambridge Univ. Press, London, 1934. Hays, C. A., and Paue, C. Y. [1] Derivation and Martingales. Springer-Verlag, Berlin. 1970. Herz, C. S. [1] "Bounded mean oscillation and related martingales." Trans. Am. Math. Soe. 193(1974) 199-215. Hida, T., and Nomoto, H. 598

[1] "Gaussian measures on the projective limit space of spheres." Proc. Japan Acad. 40(1964) 301-304. [2] "Finite dimensional approximation to band limited white noise." Nagoya Math. J. 29(1967) 211-216. Hille, E., and Phillips, R. S. [1] Functional Analysis and Semi-Groups. Am. Math. Soc. Colloq. Publ., Providence, RI, 1958. Hörmander, L. [1] "Hypoelliptic second order differential equations." Acta Math. 119(1967) 147-171. Hunt, G. A. [1] Martingales et Processus de Markov. Dunad, Paris. 1966. Hürzeler, H. E. [1] "Stochastic integration on partially ordered sets." J. Multivar. Anal. 17 (1985) 279-303. Ikeda, N., and Watanabe, S. [1] Stochastic Differential Equations and Diffusion Processes, North• Holland, Amsterdam, (2nd ed.) 1989. Ionescu Tulcea, A., and Ionescu Tulcea, C. [1] Topics in the Theory of Lifting. Springer-Verlag, Berlin. 1969. Ionescu Tulcea, C. [1] "Mesures dans les espaces produits." Att Acad. Naz. Lincei Rend. cl. Sei. Fis. Mat. Nat. 7(8)(1949/50) 208-211. Isaac, R. [1] "A proof of the martingale convergence theorem." Proc. Am. Math. Soc. 16(1965) 842-844. Isaacson, D. [1] "Stochastic integrals and derivatives." Ann. Math. Statist. 40( 1969) 1610-1616. !to, K. [1] "On a formula concerning stochastic differentials." Nagoya math. J. 3 (1951) 55-66. [2] "On stochastic differential equations." Mem. Am. Math. Soc. 4 (1951) 51pp. [3] "Stochastic differentials." Appl. Math. Opt. 1 (1975) 374-381. [4] "Multiple Wiener integral." J. Math. Soc. Japan, 3 (1951) 157- 169. Bibliography 599

Ito, K., and Watanabe, S. [1] "Transformation of Markov processes by multiplicative function• als." Ann. Inst. Fourier Grenoble, 15(1965) 13-30. [2] "Introduction to stochastic differential equations." Proc. Internat. Symp. Stoch. Diff. Eq. Kyoto, (1978) i-xxv. John, F., and Nirenberg, L. [1] "On functions of bounded mean oscillation." Comm. Pure Appl. Math. 14(1961) 785-799. Johansen, S., and Karush, J. [1] "On the semimartingale convergence." Ann. Math. Statist. 37 (1966) 680-694. Johnson, G., and Helms, L. L. [1] "Class (D) supermartingales." BuH. Am. Math. Soc. 69(1963) 59-62. Kailath, T., and Zakai, M. [1] "Absolute continuity and Radon-Nikodym derivatives for certain measures relative to Wiener measure." Ann. Math. Statist. 42 (1971) 130-140. Kakutani, S. [1] "Concrete representation of abstract (M) spaces." Ann. Math. 42(2)(1941) 994-1024. Kampe de Feriet, J. [1] "Sur un probleme d'algebre abstrait pose par la definition de la moyenne dans la theorie de la turbulence." An. Soc. Sci. Bruxelles, 63(1949) 156-172. [2] "Problemes mathematiques poses par la mecanique statistique de la turbulence." Proc. Internat. Cong. Math. 3(1954) 237-242. Kaneko, H., and Taniguchi, S. [1] A stochastic approach to Silov boundary." J. Functional Anal. 74 (1987) 415-429. Karhunen, K. [1] "Uber lineare Methoden in der Wahrscheinlichkeitsrechnung." Ann. Acad. Sci. Fenn. AI. Math. 37(1947) 3-79. Kazamaki, N. [1] "Changes of time, stochastic integrals and weak martingales." Z. Wahrs. 22(1972) 25-32. 600

[2] "Note on a stochastic integral equation." Lect. Notes Math. 258 (1972) 105-108. Kingman, J. F; C. [1] "Additive set functions and the theory ofprobability." Proc. Camb. Phil. Soc. 63(1967) 767-775. Knight, F. B. [1] "A reduction of continuous square integrable martingales to Brown• ian motion." Lect. Notes Math. 190(1971) 19-31. Kolmogorov, A. N. [1] Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer-Verlag, Berlin. 1933. [Foundations of the Theory of Probability, (Trans• lation) Chelsea Publishing Co. New York, 1956.] Krasnosel'skii, M. A., and Rutickii, Va. B. [1] Convex Functions and Orlicz Spaces. Noordhoff, Groningen, The Netherlands, 1961. Krinik, A. [1] "Diffusion processes in and likelihood ratios."in Real and Stochastic Analysis, Wiley, New York, (1986)168-210. Kunita, H. [1] Stochastic Flows and Stochastic Differential Equations. Camb. Univ. Press, London, 1990. Kunita, H., and Watanabe, S. [1] "On square integrable martingales." Nagoya Math. J. 30(1967) 209-245. Kuo, H. H. [1] "Stochastic integrals in abstract Wiener spaces." Pacific J. Math. 41(1972) 469-483. Lamb, C. W. [1] "Representation of functions as limits of martingales." Trans. Am. Math. Soc. 188(1974) 395-405. Levy, P. [1] Processus Stochastiques et Mouvement Brownien.(2nd ed.) Gauthier- Villars, Paris, 1965. Lindenstrauss, J., and Pelczynski, A. (1) "Absolutely summing operators in .cP-spaces and applications." Studia Math. 29(1968) 275-326. Lloyd, S. P. Bibliography 601

[1] "Two lifting theorems." Proc. Am. Math. Soc. 42(1974) 128-134. LOEwe, M. [1] . (3rd ed.) Van Nostrand, Princeton, NJ, 1963. Loomis, L. H. [1] Introduction to Abstract Harmonie Analysis. Van Nostrand, Princeton, NJ, 1953. Maisonneuve, B. [1] "Quelques martingales remarkables associees a une martingale continue." Publ. Inst. Statist., Univ. Paris, 17(1968) 13-27. Malliavin, P. [1] "Stochastic calculus ofvariation and hypoelliptic operators." Proc. Intern. Symp. Stoch. Diff. Eq., Kyoto, (1978) 195-263. Mallory, D. J., and Sion, M. [1] "Limits ofinverse systems ofmeasures." Ann. Inst. Fourier Greno• ble, 21(1971) 25-57. McKeanjr., H. P. [1] Stochastic Integrals. Academic Press, New York, 1969. McShane, E. J. [1] Order Preserving Maps and Integration Processes. Ann. Math. Studies, 31, Princeton Univ. Press, Princeton, NJ, 1953. [2] "Families of measures and representations of algebras of opera• tors." Trans. Am. Math. Soc. 102(1962) 328-345. [3] Stochastic Calculus and Stochastic Models. Academic Press, New York, 1974. [4] "Stochastic differential equations." J. Multivar. Anal. 5 121-177. Mertens, J.-F. [1] "Theorie des processus stochastiques generaux et surmartingales." z. Wahrs. 22( 1972) 45-68. Metivier, M. [1) Semimartingales, W. de Guyter, Berlin, 1982. Metivier, M. Pellaumail, J. [1] "On Doleans-Föllmer measure for quasimartingales." Ill. J. Math. 19(1975) 491-504. [2) Stochastic Integrals, Academic Press, New York, 1980. Meyer, P. A. [1) Probability and Potentials. Blaisdell Co., Waltham, MA, 1966. 602

[2) Martingales and Stochastic Integrals. Lect. Notes Math. 284(1972) 89pp. [3) "A decomposition theorem for supermartingales." Ill. J. Math. 6(1962) 193-205; "Uniqueness," ibid. 7(1963) 1-17. [4) "Stochastic integrals." Lect. Notes Math. 39(1967) 72-162. [5) Processus de Markov. Lect. Notes Math. 26 190pp. [6) "Une cours sur les integrales stochastiques." Lect. Notes Math. 511(1976) 245-400. [7) "Geometrie differential stochastiques." Asterisque, 131(1985) 107- 113. Millar, P. W. [1) "Transformsofstochasticprocesses." Ann. Math. Statist. 39(1968) 372-376. [2) "Martingale integrals." Trans. Am. Math. Soc. 133(1968) 145- 166. Millington, H., and Sion, M. [1) "Inverse systems of group valued-measures." Pacific J. Math. 44 637-650. Minlos, R. A. [1) "Generalized random processes and their extension to a measure." Trudy Moscow Mat. Obs<:. 8(1959) 497-518. [IMS-AMS Transla• tion No. 3(1963) 291-313.) Molchan, G. M. [1) "Some problems connected with the Brownian motion of Levy." Th. Prob. Appl. 12(1967) 682-690. Morse, M., and Transue, W. [1) "C-bimeasures and their integral extensions." Ann. Math. 64(2) (1956) 480-504. Moy, S.-C. [1] "Characterizations of conditional expectation as a transformation on function spaces." Pacific J. Math. 4(1954) 47-64. Nelson, E. [1) "Regular probability measures on function spaces." Ann. Math. 69(2)( 1959) 630-643. Neveu, J. [1) Mathematical Foundations of the Calculus of Probabilities. Holden• Day, San Francisco, CA, 1965. Bibliography 603

[2] Processus Aleatoires Gaussiens. Univ. Montreal Press, 1968. [3] "Theorie des semi-groupes de Markov." Univ. Calif. Publ. Statist. 2(1958) 319-394. Norris, J. [1] "Simplified Malliavin calculus." Lect. Notes Math. 1204(1986) 101-130. Nualart, D., and Pardoux, E. [1] "Stochastic calculus with anticipating integrands." Prob. Th. Rel. Fields, 78(1988) 535-58l. Olson, M. P. [1] "A characterization of conditional probability." Pacific J. Math. 15(1965) 971-983. Orey, S. [1] "Conditions for absolute continuity oftwo diffusions." Trans. Am. Math. Soc. 193(1974) 413-426. [2] "F-processes." Proc. Fifth Berkeley Symp. Math. Statist. Prob. 2(1958) 301-313. Paley, R. E. A. C., Wiener, N., and Zygmund, A. [1] "Notes on random functions." Math. Z. 37(1929) 647-668. Pellaumail, J. [1] "Sur l'integrale stochastique et la decomposition de Doob-Meyer." Asterisque No. 9 (1973) 1-125. Petersen, K. E. [1] Brownian Motion, Hardy Spaces and Bounded Mean Oscillation. London Math. Soc. Lect. Notes 28, 1977. Pitcher, T. S. [1] "Parameter estimation for stochastic processes." Acta Math. 112 (1964) 1-40. Postnikov, M. M. [1] The Variational Theory of Geodesics. (Translation) Dover, New York, 1983. Prokhorov, Yu. V. [1] "Convergence of random proces ses and limit theorems in proba• bility." Th. Prob. Appl. 1(1956) 157-214. [2] "The method of characteristic functionals." Proc. Fourth Berkely Symp. Math. Statist. Prob. 2(1961) 403-419. Protter, P. E. 604

[1] "On the existence, uniqueness, convergence and explosions of so• lutions of stochastic integral equations." Ann. Prob. 5(1977) 243- 26l. [2] "Right continuous solutions of systems of stochastic integral equa• tions." J. Multivar. Anal. 7(1977) 204-214. [3] Stochastic Integration and Differential Equations: A New Ap• proach. Springer-Verlag, Berlin, 1990. Rad6, T. [1] Subharmonie Functions. Chelsea, New York, 1949. Raa, K. M. [1] "On decomposition theorems of Meyer." Math. Scand. 24(1969) 66-78. [2] "Quasimartingales." ibid.24(1969) 79-92. [3] "On modification theorems." Trans. Am. Math. Soc. 167(1972) 443-450. RaoM. M. [1] "Conditional expectations and closed projections." Proc. Acad. Sei., Amsterdam, Sero A, 68(1965) 100-112. [2] "Inference in stochastic processes,I-VI." I, Th. Prob. Appl. 8( 1963) 282-298; II, Z. Wahrs. 5(1966) 317-335; III, ibid.8(1967) 49-72; IV, Sankhyä, Sero A, 36(1974) 63-120; V, ibid.37(1975) 538-549; VI, Multivariate Analysis-IV(1977) 311-334. [3] "Conditional measures and operators." J. Multivar. Anal. 5(1975) 330-413. [4] "Two characterizations of conditional probability." Proc. Am. Math. Soc. 59(1976) 75-80. [5] "Abstract nonlinear prediction and martingales." J. Mul• tivar. Anal. 1(1971) 129-157, and 9 614. [6] "Extensions of stochastic transformations." Trab. Estad. 26(1975) 473-485. [7] "Conjugate series, convergence and martingales." Rev. Roum. Math. Pures Appl. 22(1977) 219-254. [8] "Interpolation, and martingales." J. Math. Mech. 16 (1966) 543-568. [9] " analysis ofnonstationary ." Developments in Statist. Academic Press, New York,1(1978) 171-225. Bibliography 605

[10] "Stochastic proces ses and cylindrical probabilities." Sankhyä, Ser. A, 43(1981) 149-169. [11] Measure Theory and Integration. Wiley-Interscience, New York, 1987. [12] ProbabilityTheory with Applications. Academic Press, New York, 1984. [13] Conditional Measures and Applications. Marcel Dekker, New York, 1993. [14] "Non L1-bounded martingales."Lect. Notes Control Inf. Sei. 16 (1979) 527-538. [15] "Stochastic integration: a unified approach." C. R. Acad. Sei. Paris, Ser I, 314(1992) 629-633. [16] "An approach to stochastic integration." in Multivariate Analysis: Future Directions, Elsivier, New York, (1993) 347-374. [17] "L2,2-boundedness, harmonizability and filtering." Stoch. Anal. Appl. 10(1992) 323-342. [18] Stochastic Processes and Integration. Sijthoff and Noordhoff, Alphen ann den Rijn, The Netherlands, 1979. Rao, M. M., and Ren, Z. D. [1] Theory of Orlicz Spaces. Marcel Dekker, New York, 1991. Rao, M. M., and Sazonov, V. V. [1] "A projective limit theorem for probability spaces and applica• tions." Th. Prob. Appl. 38307-315. Raoult J.-P. [1] "Sur un generalization d'un theorem d'Ionescu Tulcea." C. R. Acad. Sei., Ser. A, Paris, 259(1964) 2769-2772. [2] "Limites projectives de mesures a-finites et probabilites condi• tionnelles." ibid. 260(1965) 4893-4896. Renyi, A. [1] "On a new axiomatic theory of probability." Acta Math. Sei. Hung. 6(1955) 285-333. [2] Foundations of Prob ability. Holden-Day, San Francisco, CA, 1970. Revuz, D., and Vor, M. [1] Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin, 1991. Riesz, F., and Sz.-Nagy, B. [1] Functional Analysis. F. Unger, New York, 1955. 606

Royden, H. L. [1] Real Analysis. (2nd ed.) Macmillan and Co., New York, 1969. Ryan, R. [1] "Representative sets and direct sums." Proc. Am. Math. Soc. 15(1964) 387-390. Saks, S. [1] Theory of the Integral. (2nd ed.) Hefner Publishing Co., New York, 1937. Sazonov, V. V. [1] "On perfect measures." Translations of Am. Math. Soc. 48(2)(1965) 229-254. Schatten, R. [1] A Theory of Cross Spaces. Princeton Univ. Press, Princeton, NJ, 1959. Schreiber, B. M., Sun, T. C., and Bharucha-Reid, A. T. [1] "Algebraic models for prob ability measures associated with sto• chastic processes." Trans. Am. Math. Soc. 158(1971) 93-105. Schwartz, L. [1] Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Tata Institute, Bombay, 1973. [2] "Prorobabilites cylindriques et applications radonifiantes." J. Fac. Sei. Univ. Tokyo, 18(1971) 139-286. [3] "Sur martingales reguliere a valeurs mesures et desintegrations regulieres d'une mesure." J. Anal. Math. 26(1973) 1-168. [4] Semi-Martingales sur des Varietes, et Martingales Conformes sur des Varietes Analytiques Complexes. Lect. Notes Math. 780 (1980) 132pp. Segal, I. E. [1] "Abstract probability spaces and a theorem of Kolmogoroff." Am. J. Math. 76(1954) 721-732. [2] "Equivalence of measure spaces." ibid.73 275-313. Shale, D. [1] "Invariant integration over the infinite dimensional orthogonal group and related spaces." Trans. Am. Math. Soc. 124(1966) 148- 157. Sierpinski, W. [1] Theorie des Ensembles. Warsaw, 1951. Bibliography 607

Sion, M. [1) Introduction to the Methods ofReal Analysis. Holt, Rinehart and Winston, New York, 1968. [2) A Theory of Semi-Group Valued Measures. Lect. Notes Math. 355(1973) 140pp. Snell, J. L. [1) "Applications ofmartingale system theorems." Trans. Am. Math. Soc. 73(1952) 293-312. Stein, E. M. [1) Topics in Harmonie Analysis. Princeton Univ. Press, Princeton, NJ,1970. Strassen, V. [1) "The existence of probability measures with given marginals. " Ann. Math. Statist. 36(1965) 423-439. Stricker, C. [1) "Mesure de Föllmer en theorie des quasi-martingales." Lect. Nütes Math. 485(1975) 408-419. [2) "Une caracterisation des quasi-martingales." ibid.485(1975) 420- 424. Stroock, D. W. [1) "Malliavin calculus: a functional analytic approach." J. Funet. Anal. 44(1981) 212-257. Stroock, D. W., and Varadhan, S. R. S. [1) " with continuous coefficients." Comm. Pure Appl. Math. 22345-400,479-530. [2) Multidimensional Diffusion Processes. Springer-Verlag, Berlin, 1979. Subrahmanian, R. [1) "On a generalization of martingales due to Blake." Pacific J. Math. 48(1973) 275-278. Sudderth, W. D. [1) "A'Fatou equation' für randomly stopped variables." Ann. Math. Statist. 42(1971) 2143-2146. Taniguchi, S. [1) "Malliavin's stochastic for manifold-valued Wiener functionals and its applications." Z. Wahrs. 65(1983) 269- 290. Traynor, T. 608

[1] "An elementary proof of the lifting theorem." Pacific J. Math. 53(1974) 267-272. Walsh, J. B. [1] "An introduction to stochastic partial differential equations." Lect. Notes Math. 1180(1986) 265-439. [2] "Martingales with a multi dimensional parameter and stochastic integrals in the plane." Lect. Notes Math. 1215(1986) 329-49l. Wiener, N. [1] "Differential spaee." J. Math. Phy. MIT, 2 (1923) 131-174. Wong, E., and Zakai, M. [1] "Martingales and stochastie integrals for proeesses with a multi• dimensional parameter." Z. Wahrs. 29(1974) 109-122. Wright, J. D. M. [1] "Stone algebra valued measures and integrals." Proe. London Math. Soc. 19(3)(1969) 108-122. Yaglom, A. M. [1] "Some classes of random fields in n-dimensional space related to stationary random proeesses." Th. Prob. Appl. 2(1957) 273-320. Yamasaki, Y. [1] "Projeetive limit of Haar measures on O(n)." Publ. Res. Inst. Math. Sei., Kyoto Univ. 8(1973) 141-149. Yeh, J. [1] "Wiener measure in a spaee offunctions of two variables." Trans. Am. Math. Soe. 95(1960) 433-450. [2] "Two parameter stoehastic differential equations." in Real and Stochastic Analysis, Wiley, New York, (1986) 249-344. Ylinen, K. [1] "On vector bimeasures." Ann. Mat. Pura Appl. 117(4)(1978) 115-138. Vor, M. [1] "Existenee et unicite de diffusion a valuers dans un spaee de Hilbert." Ann. Inst. H. Poineare, 10(1974) 55-88. [2] "Sur quelques approximations d'integrales stoehastiques." Leet. Notes Math. 581(1977) 518-528. Yosida, K., and Kakutani, S. [1] "Operator theoretieal treatment of Markoff's proeesses and mean ergodie theorem." Ann. Math. 42 (1941) 188-228. Bibliography 609

Zaanen, A. C. [1] "The Radon-Nikodym theorem." Proc. Aead. Sei., Amsterdam, Sero A 64(1961) 157-187. Zygmund, A. [1] Trigonometrie Series. Cambrige Univ. Press, London, 1958. Notation Index

Chapter I

(n, ~,P)-Probability triple, 1 (lR[C] denotes real [complex] field) [a, b), (a, b], [a, b]-the usual half-open and closed intervals E(X)-expectation of X, 8 C BS=Cauchy-Buniakowski-Schwarz, 8 iff = if and only if, 9 lR T-space of real functions on a set T, 11 R-Borel a-algebra of lR, 13 {(na,~a,Pa,gaß)a<ß: Ct,ß in D}-projective system, 17 X - usually a Banach space, X* its dual, X' its algebraic dual, 34 (i, X, B)-abstract Wiener triple, 41

Chapter 11

EB(X)-conditional expectation of X relative to a a-algebra ß, 64 PB = PIß, restriction of P to ß, 63 p B _ conditional probability relative to ß, 64 ~(Y) or ~y- a-algebra generated by the given function Y, 75 V(I\)-maximum (minimum) operator, 105 (n, ~,ß, P(·I·))-Renyi's conditonal measure space, 104

E(Xtn+1 IXt1 , ... ,Xtn )-conditional expectation, 111 LP(n,Fs,p) = LP(Fs ), 111 .€g+ A = log+ A = log A for A > 1, = 0 for A ::; 1, 114

611 612 Notation Index

~v;, -Radon-Nikodym derivative of the P-continuous part of 1I, 142

Chapter III

CP, LP, C=, L=-Lebesgue spaces, 177 p(.) - a lifting map, 177 M= - space of real bounded measurable functions, 177 Ab.B - symmetrie difference of sets A and B, 178 Ta - evaluation functional, 183

Chapter IV

ß(T) = ßT - u-algebra of events prior to the optional T, 242 O:T(Xd = XT/\t - stopped process at T, 258 (}T(Xd = XT+t - translated process by T, 258 NA - nucleus operator, 324

Chapter V

[T1,T2 1 = ((t,w): T1(w):S: t:s: T2 (w)} - stochastic interval, 361 (X, Y) - quadratic covaration (continuous case), 376 [X, Yj - quadratic covariation (general case), 385 M,Mloc,Mc,A+ - classes ofprocesses, 377 LP(A),CP((X)) - spaces ofprocesses, 391 X t = Y 0 Tt or Yi = X 0 Tt- 1 - strict time change, 406 ß(~+) - Borel u-algebra of ~+, 4132

Chapter VI

Xi - the continuous part of a martingale, 448 S(D',O) - O-measurable simple functions, 462 Co - the Banach space of numerical sequences converging to zero, 470 b.Xt = X t - X t- - the jump of the process at t, 474 ey = {Et , tEl}, E t = B(D, F t , Y)-envelope of the Banach spaces of Y-valued Ft-measurable bounded functions, 475 Notation Index 613 s(n, Ey) - normed vector space of "vector fields", 476 L'j;(P) - Orlicz space of Z-valued functions, 477 Mi(P) - subspace of L'j;(P) determined by simple functions, 477 x 02 = x ® x, tensor product of x, 479 [X] - quadratic variation of vector valued X, 480 (A)ij - the (i,j)th - element of an m-by-n matrix, 492 Xt - derived process of X t , 494 n a=(al, ... ,an),ak = TI a7 i ,k=(k1 , ... ,kn ),lkl=k1 +···+kn ,512 i=1 Co (IR. n, IR.) - the space of real continuous functions on IR.n vanishing at =, 528

Chapter VII

{HP, 11 . IIp} - Hardy space oflocal martingales, 540 {BMO, II·IIB} - the space of bounded mean oscillation for martingales, 540 H(Z) - dass of complex holomorphic processes, 544 Tx(M) - tangent space at x of a manifold M, 546 C( M) - the set of Coo - vector fields, 547 r~j - Christoffel symbol, 550 T~(M) = Tx(M) + iTx(M) - complexified tangent spae, 553 GL(M) - orthogonal frame bundle, 556 O(H) - bundle of orthogonal frames, 557

[Ai, A j ] = A;Aj - AjA;, the Lie bracket, 560 Ft - augmented filtration, 571 P* - augmented predictable a-algebra, 571 C6, C60 - dasses of two parameter processes, 576 Ms,.Afs - orthogonal complementary dasses of two parameter martingales, 577 yaX - stochastic differential form, 578 X (81,82) - modified Brownian sheet, 580 Author Index 615

Author Index

Adams, C.R, 569 Chung, K.L., 194, 257, 330, 380, Alloin, C., 299 424 Ambrose, W., 168 Clarkson, J.A., 569 Andersen, E.S., 142, 146, 148, 163 Coddington, E.A., 488 Anderson, RF., 527 Cornea, A., 345, 440 Auslander, L., 585 Courrege, P., 389, 459, 479 Austin, D.G., 294, 331, 385 Cramer, H., 389, 459, 479 Cuculescu, 1., 423, 430, 431, 443 Bartle, RG., 485 Bell, D.R., 586 Dalecky, Ju.L., 535, 562, 586 Belopol'skaya, Ya.1., 562, 586 Dambis, K.E., 403, 411, 442 Bharucha-Reid, A.T., 60 Darling, RW.R., 529 Bichteler, K., 475 Davis, B.J., 300, 310 Bismut, J.-M., 585 Davis, M., 26 Blackwell, D., 259 de la Vallee Poussin, C.J., 79 Blake, L.H., 161 Dellacherie, C." 363, 366, 373, 375, Blumenthai, RM., 509 475,509 Bochner, S., 10, 19, 20, 59, 112, Dinculeanu, N., 21, 92, 96, 102, 151, 197, 330, 445, 456, 536 155, 162, 311, 443, 470, 478 Bonami, A., 527 Doleans-Dade, C., 318, 363, 364, Borchers, D.R, 496, 522, 536 366,439,441,442,530,535,571 Bourbaki, N., 21, 31, 281 Doob, J.L., 55, 75, 104, 115, 116, Brennan, M.D., 536, 569, 571, 122, 126, 134, 157, 163, 168, 574, 584, 586 171,191,194,197,230,257, Brooks, J.O., 443 263, 291, 322, 330, 380, 424, Burkholder, D.L., 310 436, 442, 521 Dozzi, M., 574, 586 Cabaiia, E.M., 535 Dubins, L.E., 163, 413, 442 Dugundji, J., 229 Cairoli, R, 564, 569, 570, 578, 586 Carmona, RA., 528 Dunford, N., 92, 95, 96, 98, 99, 101, Cartan, H., 513 106, 147, 151, 159, 178, 213, Cartier, P., 482 214, 226, 279, 344, 389, 464, Chacon, RV., 442 470,534 Dym, H., 512, 536 Chang, D.K., 585 Dynkin, E.B., 75, 151, 509 Choksi, J.R, 20, 23, 60, 220, 223, 231 Choquet, G., 328 Eberlein, W.F., 213 Chow, Y.S., 70, 146, 157, 162, Edgar, G.A., 252, 442 211, 229, 231, 269, 281, Emery, M., 553, 585 285, 330, 331 616 Author Index

Fefferman, C.L., 301, 331 Hürzeler, H.E., 586 Feldman, J., 58 FeIler, W., 505 Ikeda, N., 502, 529, 537, 559, Fillmore, P.A., 228 562, 583, 586 Finetti, B.de, 1 Ionescu Tulcea, A., 176, 185, Finlayson, H.C., 46 218,230 Fisk, D.L., 364, 418, 434, 442, 443 Ionescu Tulcea, C.T., 176, 185, Flanders, H., 585 218, 224, 225, 226, 230, 231 Föllmer, H., 369, 442, 543, 585 Isaac, R., 134 Freedman, D., 259 Isaacson, D., 439 Freidlin, M., 587 rtö, K., 36, 126, 334, 345, 419, 437, Fremlin, D.H., 180 442, 443, 521, 531

GangoIli, R., 482 Jensen, B., 142, 146, 148, 154, 163 Garsia, A.M., 300, 310, 312, 331 Johansen, S., 163 Gel'fand, I.M., 35, 36, 39 John, F., 312, 331 Getoor, R.K., 509, 541, 542, 585 Johnson, G., 263, 324, 330 Gikhman, I.L., 231 Girsanov, LV., 439 Girshick, M.A., 162 Kailath, T., 439 Gnedenko, B.V., 102 Kakutani, S., 537 Goldberg, S., 585 Kampe de Feriet, J., 162 Goldstein, J.A., 489 , 524 , 525 , Kaneko, H., 583 536, 537 Karhunen, K., 389 Gould, G.G., 389 Karoui, N., 527 Green, M.L., 586 Karush, J., 163 Gross, L., 33, 41, 46, 58, 60 Kazamaki, N., 432, 535 Guggenheim, H.W., 585 Kingman, J.F.C., 55 Gundy, R.F., 288, 310, 331 Knight, F.B., 411 Kolmogorov, A.N., 1, 10, 15, 16,35, 59, 102, 104, 112, 116, Hajek, J., 58, 116, 119 121, 154, 162, 197, 200, Halmos, P.R., 2, 11, 24, 55, 226 224, 389, 459,479 Hankel, H., 195 Krasnosel'skii, M.A., 156 Hanner, 0., 176 Krinik, A., 527, 535 Hardy, G.H., 69 Kunita, H., 375, 384, 405, 442, 443, Hayes, C.A., 229, 231 529,535,537,582,587 Helms, L.L., 263, 324, 330 Kuo, H.H., 535 Herz, C.S., 331 Hida, T., 47 Hille, E., 486 Lamb, C.W., 289, 329, 331 Hörmander, L., 559, 585 Levinson, N., 488 Hunt, G.A., 242, 322, 330 Levy, P., 52, 55, 158, 163, 403 , 404 , Author Index 617

482, 486 Pardoux, E., 439,532 Licea, G., 440 Panc, C.Y., 229, 231 Littlewood, J.E., 69, 291 Pellaurnail, J., 443, 503, 506 Lohe, M., 146, 202 Petersen, K.E., 331 Loornis, L.H., 88, 423 Phillips, RS., 496 Luzin, N., 326 Pitcher, T.S., 162 Poincare, H., 5 P61ya, G., 69 MacKenzie, RE., 585 Postnikov, M.M., 585 Maisonneuve, B., 439, 530 Priouret, P., 257, 323 Malliavin, P., 560, 561, 585 Prokhorov, Yu. V., 31, 32, 42, 281 Mallory, D.J., 55, 220 Protter, P.E., 536 Marcinkiewicz, J., 122, 123 Maruyarna, G., 519 McKean, Jr., H.P., 436, 443 Rad6, T., 133, 431 McShane, E.J., 97, 231, 443, Rao, K.M., 230, 369, 370, 433, 442 453, 454, 455, 456, 536 Reinhard, H., 527 Mertens, J.-F., 230, 440 Ren, Z.D., 156, 469, 470, 566 Metivier, M., 443, 503, 536 Renyi, A., 1, 104, 105, 116, 119 Meyer, P.-A., 126, 128, 227, 230, Riesz, F., 126, 498 259, 267, 328, 330, 348, Royden, H.L., 2, 21, 81, 197 356, 363, 373, 390, 430, Roynette, B., 527 432, 442, 443, 509, 535, 585 Rutickii, Ya.B., 156 Millar, P.W., 330, 441 Ryan, R, 179,231 Millington, H., 19, 57 Minios, RA., 35, 39, 46 Saks, S., 326 Molchan, G.M., 486 Savage, L.J., 1, 162 Morse, M., 585 Sazonov, V.V., 60, 102, 103, 218, Moy, S.-C., 155, 162 219,231 Schatten, R, 478, 480 Nelson, E., 171, 231 Schreiber, B.M., 60 Neumann, J. von, 228 Schwartz, J.T., 92, 95, 96, 98, 99, Neveu, J., 202, 224, 231, 291, 431 101, 106, 147, 151, 159, 178, Nirenberg, L., 312, 331 213, 214, 226, 279, 344, 389, Nornoto, H., 47 464, 470, 534 Norris, J., 562 Schwartz, L., 35,66,231, 281, 583, Nualart, D., 439, 528, 532 585 Schwarz, G., 403" 442 Segal, LE., 36, 60, 231 Olson, M.P., 163 Shale, D., 50 Orey, S., 432, 439, 502 Sharpe, M.J., 541, 542, 585 Sierpinski, W., 326, 328 Paley, RE.A.C., 291, 459 Sion, M., 2, 19, 55, 57, 176, 618 Author Index

178, 220, 230 Zaanen, A.C., 231 Skorokhod, A.V., 231, 531 Zakai, M., 439, 576, 586 Snell, J.L., 269, 330 Zygmund, A., 69, 123, 291, 459 Stein, E.M., 291, 301, 331 Steinhaus, H., 154 Strassen, V., 162 Stratonovich, R.L., 418, 443 Stricker, C., 369, 432 Stroock, D.W., 527, 530, 535, 585 Subrahmanian, R., 161 Suchest on, L., 252, 442 Sudderth, W., 321 Sun, T.-C., 60 Sz.-Nagy, B., 498

Taniguchi, S., 562, 583 Transue, W., 585 Traynor, T., 176,178 Thlcea, (see Ionescu Thlcea)

Varadhan, S.R.S., 529, 530, 535 Vilenkin, N.Ya., 35, 39 Ville, J., 163

Walsh, J.B., 569, 570, 575,578, 580, 584, 586, 587 Watanabe, S., 126, 334, 345, 375, 384, 405, 419, 442, 502, 529, 536, 559, 562, 583, 586 Whitney, H., 547, 549 Wiener, N., 45, 46, 389, 443, 459 Wong, E., 576, 586 Wright, J.D.M., 97

Yaglom, A.M., 486 Yamasaki, Y., 50 Yeh, J., 46, 579, 586 Ylinen, K., 585 Yor, M., 422,443,527 Yosida, K.,. 537 Yushkevich, A.A., 509 619

SUBJECT INDEX A Brownian flow, 528 funetional, 576 motion (and density), 52 A-analytie, 325 martingale eharacterization, 406 absolutely eontinuous (At) in a manifold, 551 relative to (Bt ), 382 Brownian sheet, 575 abstract , 1, 41 bundle of orthogonal frames, 557 abstract Wiener spaee, 35, 526 unitary frames, 558 aeeessible time, 361 aeeessible a-algebra, 362 adapted to a a-algebra, 110 C adjunction proeedure, 98 ca,dlag, 230 adjoint operator, 81 canonical (= function space) admissible translate, 161 representation, 16 affine eonneetion, 555 Caratheodory generated measure, 54 a-stoehastie integral, 422 Cauchy distribution family, 52 almost separable projeetive Cauehy-Riemann equations, 544 system, 219 CBS-inequality, 8 amart (=asymptotie martingale), Cebysev inequality, 119 441 Chapman-Kolmogorov equation 505 analytie manifold, 545 characteristic function, 6 analytie (=A or Soustin) properties of, 51 operation, 325 characterization of projective limit A-(quasi, sub-) martingale, 570 measures, 29 approximate martingale, 159 Brownian motion, 406 A-proeess, 570 chart, 546 are length function, 548 Christoffel symbols, 550 assoeiated projeetive system, 279 dass (D), (DL), 262 asymptotie martingale, 441 CP-manifold, 545 atlas, 546 compatibility condition, 3 averaging identity, 84 generalized, 17 averaging property, 69 compensator, 387 eomplex Brownian motion, 539 complex derivative of a process, 544 B complex martingale, 540 baekward integral, 422 conditional convergence theorem, 67 Banaeh contraction mapping, 534 distribution, 153 base of a martingale, 20 formulas for, 153 BMO, 233, 300 expectation, 62, 64 Borel-Cantelli lemma, 298 characterization of, 85 boundedly a-eomplete lattiee, 97 generalized, 151 b-quadratie variation, 549 independence, 152, 574 Jensen inequality, 70 620 Subject Index

probability, 62, 64 (continuous case), 233, 338 as vector measure, 92 drift coefficient, 519 characterization of, 105 Dunford-Schwartz integral, 95 regularity of, 92 conditionally negative E definite, 483 c-(stochastic) partition, 397 conformal basis, 543 event, 1 martingale, 540 evolution equation, 506 semimartingale, 540 expectation, 8 on a manifold, 584 conjugate martingale, 540 connection, 550 F coordinate neighborhood, 546 Feller semigroup, 507 cotangent space, 547 process, 507 control measure, 95 F -quasimartingale, 583 covariance function, 8 fine covering, 211 covariant differentiation, 555 essential covering, 211 cross section,176 finite subset property, 179 local, 176 fixed point of discontinuity, 173 cylindrical probability, 36 fundamental theorem of (stochastic) calculus, 437 D debut, 258 G degenerate elliptic operator as generalized Jordan generator, 523 decomposition, 370 d-sequence, 290 random process, 36 tl-splitting, 289 generator of a potential, 344 density function, 7 geodesic, 555 diffeomorphic, 547 graph (of an optional), 362 diffusion coefficient, 519 grid, 569 process, 519 direct sum property, 179 H directed set, 16 hermitian metric, 553 discontinuity, point, 195 Hessian, 549 second kind, 194 holomorphic complex process, 543 distribution function, 2 manifold, 546 joint, 3 real process, 578 Gaussian, 5 horizontal lift , 557 Poisson,6 space, 557 distribution al solution, 580 H-process, 570 Doob decomposition (discrete H-quasimartingale, 570 case) , 126 hypoelliptic operator, 560 Doob-Meyer decomposition Subjeet Index 621

strietly, 179

I image probability, 2 i-martingale, 571 M inaeeessible time, 361 Malliavin eovarianee, 561 independent inerements, 9 Markov ehain, 504 mutually,9 proeess, 503 o--algebras, 77 strong,508 indicator funetion, 8 time reversed, 504 indistinguishable, 392 martingale, 20, 110 for MeShave differenee (= inerement), 295 integrals, 455 loeal, 334 for symmetrie integral, 418 regular, 269 Itö's formula, (generalized), 395 semiregular, 269 for semimartingales, 448 wide sense, 111 Itö's integral, 391, 435 martingale eonvergenee, 140, 148 MeShane's integral, 454 K mean funetion, 8 Kähler manifold, 555 mean square derivative, 489 Kählerian metrie, 554 measurable proeess, 171 K.6t-eondition, 453 measurable , 40 K -independenee, 154 metrie tensor, 548 Kolmogorov's inequality, 118 modelled after ]Rd, 546 Kroneeker's lemma, 157 modifieation of a proeess, 166 moving point of diseontinuity, 175 M-proeess, 567 L multiplieation table (for Langevin type equation, 487 stoehastie differentials), 420 Laplaee-Beltrami operator, 551 Levy-Brownian motion, 482 L2 ,2-boundedness, 457 N Lie derivative, 550 natural (= predictable) proeess, 344 lifting, linear, 177 nearly a K.8t-eondition, 453 lifting map, 177 nonantieipative integrand, 471 set, 179 normalization of H-proeess, 572 topology, 228 nucleus, 324 Dill ,'P2-boundedness, 461 veetor ease, 476 LP'P-boundedness, 461 o one sided Fubini's theorem, 108 loeal eoordinate, 546 operation (A), 324 loeal martingale, 334 optimal stopping rule, 299 loealizability, 179 622 Subject Index optional (= ), 234 generalized, 440 optional a-algebra, 362 of L2,2-bounded proeesses, sampling process, 239 472 (= well-measurable) process, 395 quasi-Ieft-eontinuous, 361 order eontinuous a-algebras, 208 quasimartingale, 364 Ornstein-Uhlenbeek sheet, 583 inequalities for, 434 orthogonal frames, 556 bundle, 556 R orthogonal inerements, 10 random linear mapping, 36 orthonormal basis of martingales, (field) funetion, 2 543 proeess, 1 Sehwartz distribution, 36 p variable, 1 parallel displaeement, 555 raw inereasing process, 387 perfeet probability, 102 realization (of a process), 2 <1>- bounded, 584 reduee (strongly) a proeess, 436 point diseontinuity, 195 regular, Baire, 21 Polish spaee, 201 Borel21 potential, 130 part of a set, 289 predietable (multivariate), 568 rieh a-algebra, 70 proeess, 337 Riemannian metrie, 547 a-algebra, 362 Riesz deeomposition, 126, 130 time, 361 right (left) eontinuous filtration, 252 pervisible (= predictable) time, 361 S prineiple bundle, 557 sampie function, 2 P-rieh a-algebra, 70 semieompaet, 325 proeess, canonieal semimartingale (loeal), 364 representation, 16 ingegrals, 448 function spaee, 16 semiring, 11 progressively measurable, 196 semipotential, 371 set, 258 semivariation, 94 projeetive limit of mappings, 29 separability of proeesses, 166 projeetive limit of (signed, sequential maximality (s.m.), 19 veetor) measures, 19 set martingale, 20 system, 17 S-independenee, 154 representation, 24 simple harmonie oseillator, 487 topologieal, 17 Skorokhod's integral, 531 purely finitely additive, 147 Sobolev type norm, 467 Souslin operation, 324 Q speetral measure, 423 quadratie variation, 318 s-regular potential,355 Subject Index 623 stable distribution, 52 tr= trace, 478 standard filtration, 334 trajectory, 2 stochastic base (= filtration transition prob ability function, 506 of adapted-process), 20 stationary, 506 fiow, 528 stochastic differential equation U (first order), 457 uniform integrability, 115 forms,578 terminal, 115, 213 * -representation, 423 unitary frame bundle, 557 stochastic integral, 389 universal separating set, 168 integrator, 462 universally measurable set, 218 vector,477 upcrossings inequality, 158 interval, 361, 392 measure, 457 partition, 397 V processes, 1 variance function, 8 types, 7 variation of a measure, 94 strictly stationary, 7 vector field, 547 weakly stationary, 8 Vitali condition "Vo", 211 stoppinig time ( = optional), 234 V-process, 570 process, 234 V-(quasi, sub) martingale, 570 Stratonovich's integral, 418 V-strong submartingale, 571 strong martingale, 571 law of large numbers, 121 W type (p,p), 125 weak distribution, 36 strongly normal, 539 martingale, 432, 570 sub (super) martingale, 111 weak solution, 580 semiregular, 269 type (1,1), 125 subprocess, 567 unit,85 sub (super) projective system, 280 weakly harmonizable process, 459 super process, 567 well measurable (= optional) pro• symmetrie integral, 418 cess, 391 white noise, 487 T Wiener-Brownian motion, 563 tangent bundle, 547 Wiener chaos, 531 space, 546 Wiener distribution, 52 vectors, 546 thick set, 53 y threads, 17 Young function, generalized, 460 time change transformation, 405 times of discontinuity, 361 torsionfree connection, 550 totally inaccessible time, 361 Other and Its Applications titles of interest:

P.M. Alberti and A. Uhlmann: Stochasticity and Partial Order. Doubly Stochastic Maps and Unitary . 1982, 128 pp. ISBN 90-277-1350-2 A.V. Skorohod: Random Linear Operators. 1983,216 pp. ISBN 90-277-1669-2 I.M. Stancu-Minasian: Stochastic Programming with Multiple Objective Functions. 1985,352 pp. ISBN 90-277-1714-1 L. Amold and P. Kotelenez (eds.): Stochastic Space-Time Models and Limit Theorems. 1985,280 pp. ISBN 90-277-2038-X Y. Ben-Haim: The Assay of Spatially Random Material. 1985,336 pp. ISBN 90-277-2066-5 A. Pazman: Foundations ofOptimum Experimental Design. 1986,248 pp. ISBN 90-277-1865-2 P. Kree and C. Soize: Mathematics of Random Phenomena. Random Vibrations of Mechanical Structures. 1986,456 pp. ISBN 90-277-2355-9 Y. Sakamoto, M. Ishiguro and G. Kitagawa: Akaike Information Criterion Statis• tics. 1986,312 pp. ISBN 90-277-2253-6 GJ. Szekely: Paradoxes in Probability Theory and . 1987, 264 pp. ISBN 90-277-1899-7 0.1. Aven, E.G. Coffman (Jr.) and Y.A. Kogan: Stochastic Analysis of Computer Storage. 1987,264 pp. ISBN 90-277-2515-2 N.N. Vakhania, V.I. Tarieladze and S.A. Chobanyan: Probability Distributions on Banach Spaces. 1987,512 pp. ISBN 90-277-2496-2 A.V. Skorohod: Stochastic Equationsfor Complex Systems. 1987,196 pp. ISBN 90-277-2408-3 S. Albeverio, Ph. Blanchard, M. Hazewinkel and L. Streit (eds.): Stochastic Processes in Physics and Engineering. 1988,430 pp. ISBN 90-277-2659-0 A. Liemant, K. Matthes and A. Wakolbinger: Equilibrium Distributions of Branching Processes. 1988,240 pp. ISBN 90-277-2774-0 G. Adomian: Nonlinear Stochastic Systems Theory and Applications to Physics. 1988,244 pp. ISBN 90-277-2525-X J. Stoyanov, O. Mirazchiiski, Z. Ignatov and M. Tanushev: Exercise Manual in Probability Theory. 1988,368 pp. ISBN 90-277-2687-6 E.A. Nadaraya: Nonparametric Estimation of Probability Densities and Regression Curves. 1988,224 pp. ISBN 90-277-2757-0 H. Akaike and T. Nakagawa: Statistical Analysis and Control of Dynamic Systems. 1998,224 pp. ISBN 90-277-2786-4 Other Mathematics and Its Applications titles of interest:

A.V. Ivanov and N.N. Leonenko: Statistical Analysis of Random Fields. 1989,256 pp. ISBN 90-277-2800-3 V. Paulauskas and A. Rackauskas: Approximation Theory in the . Exact Results in Banach Spaces. 1989, 176 pp. ISBN 90-277-2825-9 R.Sh. Liptser and A.N. Shiryayev: Theory of Martingales. 1989,808 pp. ISBN 0-7923-0395-4 S.M. Ermakov, V.V. Nekrutkin and A.S. Sipin: Random Processes for Classical Equations of Mathematical Physics. 1989, 304 pp. ISBN 0-7923-0036-X G. Constantin and I. Istratescu: Elements of Probabilistic Analysis and Applica• tions. 1989,488 pp. ISBN 90-277-2838-0 S. Albeverio, Ph. Blanchard and D. Testard (eds.): Stochastics, Algebra and Analysis in Classical and Quantum Dynamies. 1990, 264 pp. ISBN 0-7923-0637-6 Ya.I. Belopolskaya and Yu.L. Dalecky: Stochastic Equations and Differential Geometry. 1990,288 pp. ISBN 90-277-2807-0 A.V. Gheorghe: Decision Processes in Dynamic Probabilistic Systems. 1990,372 pp. ISBN 0-7923-0544-2 V.L. Girko: Theory of Random Determinants. 1990, 702 pp. ISBN 0-7923-0233-8 S. Albeverio, PH. Blanchard and L. Streit: Stochastic Processes and their Applica• tions in Mathematics and Physics. 1990,416 pp. ISBN 0-9023-0894-8 B.L. Rozovskii: Stochastic Evolution Systems. Linear Theory and Applications to Non-linear Filtering. 1990,330 pp. ISBN 0-7923-0037-8 A.D. Wentzell: Limit Theorems on Large Deviations for Markov . 1990, 192 pp. ISBN 0-7923-0143-9 K. Sobczyk: Stochastic Differential Equations. Applications in Physics, Engineer• ing and Mechanics. 1991,410 pp. ISBN 0-7923-0339-3 G. Dallaglio, S. Kotz and G. Salinetti: Distributions with Given Marginals. 1991, 300 pp. ISBN 0-7923-1156-6 A.V. Skorohod: Random Processes with Independent Increments. 1991,280 pp. ISBN 0-7923-0340-7 L. Saulis and V.A. Statulevicius: Limit Theorems for Large Deviations. 1991,232 pp. ISBN 0-7923-1475-1 A.N. Shiryaev (ed.): Selected Works of A.N. Kolmogorov, Vol. 2: Probability Theory and Mathematical Statistics. 1992,598 pp. ISBN 90-277-2795-X Yu.1. Neimark and P.S. Landa: Stochastic and Chaotic Oscillations. 1992,502 pp. ISBN 0-7923-1530-8 Other Mathematics and Its Applications titles of interest:

Y. Sakamoto: Categorical Data Analysis by Ale. 1992, 260 pp. ISBN 0-7923-1429-8 Lin Zhengyan and Lu Zhuarong: Strong Limit Theorems. 1992, 200 pp. ISBN 0-7923-1798-0 J. Galambos and I. Katai (eds.): Probability Theory and Applications. 1992, 350 pp. ISBN 0-7923-1922-2 N. Bellomo, Z. Brzezniak and L.M. de Socio: Nonlinear Stochastic Evolution Problems in Applied Sciences. 1992, 220 pp. ISBN 0-7923-2042-5 A.K. Gupta and T. Varga: Elliptically Contoured Models in Statistics. 1993, 328 pp. ISBN 0-7923-2115-4 B.E. Brodsky and B.S. Darkhovsky: Nonparametrie Methods in Change-Point Problems. 1993,210 pp. ISBN 0-7923-2122-7 V.G. Voinov and M.S. Nikulin: Unbiased Estimators and Their Applications. Volume 1: Univariate Case. 1993,522 pp. ISBN 0-7923-2382-3 V.S. Koroljuk and Yu.V. Borovskich: Theory oJU-Statistics. 1993,552 pp. ISBN 0-7923-2608-3 A.P. Godbole and S.G. Papastavridis (eds.): Runs and Patterns in Probability: Selected Papers: 1994,358 pp. ISBN 0-7923-2834-5 Yu. Kutoyants: Identification oJ Dynamical Systems with Small Noise. 1994, 298 pp. ISBN 0-7923-3053-6 M.A. Lifshits: Gaussian Random Functions. 1995,346 pp. ISBN 0-7923-3385-3 M.M. Rao: Stochastic Processes: General Theory. 1995,635 pp. ISBN 0-7923-3725-5