Absolute continuity and singularity of locally absolutely continuous probability distributions. II

Yu. M. Kabanov   ; R. Sh. Liptser   ; A. N. Shiryaev

Sbornik. Mathematics, Tome 36 (1980) no. 1, pp. 31-58

Résumé

The second part of this study deals with criteria for the absolute continuity of measures for various classes of random processes, starting out from the general results in Part I. We consider processes with independent increments, semimartingales, multivariate point processes, Gaussian processes, Markov chains, and processes with a countable number of states. Bibliography: 33 titles.

Bibliographie

[1] Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev, “K voprosu ob absolyutnoi nepreryvnosti i singulyarnosti veroyatnostnykh mer”, Matem. sb., 104(146) (1977), 227–247 | MR | Zbl

[2] S. Kakutani, “On equivalence of infinite product measures”, Ann. of Math. (2), 49:1 (1948), 214–224 | DOI | MR | Zbl

[3] P. A. Meyer, “Un cours sur les intégrates stochastiques”, Lecture Notes in Math., 511, 1976, 245–400 | MR | Zbl

[4] A. V. Skorokhod, Sluchainye protsessy s nezavisimymi prirascheniyami, izd-vo “Nauka”, Moskva, 1964 | MR

[5] I. I. Gikhman, A. V. Skorokhod, Vvedenie v teoriyu sluchainykh protsessov, izd-vo “Nauka”, Moskva, 1965 | MR

[6] T. T. Kadota, L. A. Shepp, “Conditions for the absolute continuity between a certain pair of probability measures”, Z. W.-theorie, 16 (1970), 250–260 | MR | Zbl

[7] I. V. Girsanov, “O preobrazovanii odnogo klassa sluchainykh protsessov s pomoschyu absolyutno nepreryvnoi zameny mery”, Teoriya veroyatn., V (1960), 314–330 | MR

[8] R. Sh. Liptser, A. N. Shiryaev, Statistika sluchainykh protsessov, izd-vo “Nauka”, Moskva, 1974 | MR

[9] J. Jacod, “A general theorem of representation for martingales”, Froc. of Symposia in Pure Math., 31 (1977), 37–53 | MR

[10] I. A. Ibragimov, Yu. A. Rozanov, Gaussovskie sluchainye protsessy, izd-vo “Nauka”, Moskva, 1970 | MR

[11] L. A. Shepp, “Radon–Nikodym derivative of Gaussian measures”, AMS, 37 (1966), 321–354 | MR | Zbl

[12] T. Kailath, “The structure of Radon–Nikodym derivatives with respect io Wiener measure and related measures”, AMS, 42 (1971), 1054–1067 | MR | Zbl

[13] Yu. V. Prokhorov, “Skhodimost sluchainykh protsessov i predelnye teoremy teorii veroyatnostei”, Teoriya veroyatn., I (1956), 177–238

[14] M. P. Ershov, “Ob absolyutnoi nepreryvnosti mer, otvechayuschikh protsessam diffuzionnogo tipa”, Teoriya veroyatn., XVII (1972), 173–178

[15] I. I. Gikhman, A. V. Skorokhod, “O plotnostyakh veroyatnostnykh mer v funktsionalnykh prostranstvakh”, Uspekhi matem. nauk, XXI:6(132) (1966), 83–152

[16] A. Segall, T. Kailath, Radon–Nikodym derivatives with respect to measures induced by discontinuous independent – increments process, 3 (1976), 449–464 | MR

[17] T. Kailath, M. Zakai, “Absolute continuity and Radon–Nikodym derivatives for certain measures relative to Wiener measure”, Ann. Math. Statist., 42 (1971), 130–140 | DOI | MR | Zbl

[18] P. Sh. Liptser, A. N. Shiryaev, “Ob absolyutnoi nepreryvnosti mer, sootvetstvuyuschikh protsessam diffuzionnogo tipa, otnositelno vinerovskoi”, Izv. AN SSSR, seriya matem., 36:4 (1972), 847–889 | Zbl

[19] B. I. Grigelionis, “O strukture plotnostei mer, sootvetstvuyuschikh sluchainym protsessam”, Lit. matem. sb., XIII:1 (1973), 71–78 | MR

[20] S. Orey, “Conditions for the absolute continuity of two diffusions”, Trans. Amer. Math. Soc., 193 (1974), 413–426 | DOI | MR | Zbl

[21] Yu. A. Rozanov, Teoriya obnovlyayuschikh protsessov, izd-vo “Nauka”, Moskva, 1974 | MR

[22] J. Jacod, J. Memin, “Charactéristiques locales et conditions de continuité absolue pour les sémimartingales”, Z. W.-theorie, 35 (1976), 1–37 | MR | Zbl

[23] J. Jacod, “Multivariate point processes: predictable projection, Radon–Nikodym derivatives, representation of martingales”, Z. W.-theorie, 31 (1975), 235–253 | MR | Zbl

[24] B. I. Grigelionis, “Sluchainye tochechnye protsessy i martingaly”, Lit. matem. sb., XV:3 (1975), 101–114 | MR

[25] R. S. Lipcer, A. N. Shiryaev, Statistics of random processes, v. I, II, Springer-Verlag, New York–Heidelberg–Berlin, 1978

[26] Yu. M. Kabanov, R. Sh. Liptser, A. N. Shiryaev, “Martingalnye metody v teorii tochechnykh protsessov”, Trudy shkoly-seminara po teorii sluchainykh protsessov, chast II (Druskininkai, 25–30 noyabrya 1974 g.), Vilnyus, 1975, 269–354 | MR

[27] Yu. M. Kabanov, R. S. Lipcer, A. N. Shiryaev, “Criteria of absolute continuity of measures corresponding to multivariate point processes”, Proc. of the Third Japan–USSR Symposium on Probab. Theory, Lecture Notes in Math., 550, 1976, 232–252 | MR | Zbl

[28] P. A. Meier, Veroyatnost i potentsialy, izd-vo “Mir”, Moskva, 1973

[29] R. D. LePage, V. Mandrekar, “Equivalence–singularity dichotomies from zero-one laws”, Proc. Amer. Math. Soc., 31 (1972), 251–254 | DOI | MR

[30] R. D. LePage, V. Mandrekar, “On likelihood ratios of measures given by Markov chains”, Proc. Amer. Math. Soc., 52 (1975), 377–380 | DOI | MR | Zbl

[31] H. Kunita, “Absolute continuity of Markov processes”, Lecture Notes in Math., 511, 1976, 44–77 | MR | Zbl

[32] H. J. Engelbert, A. N. Shiryaev, “On absolute continuity and singularity of probability measures”, Mathematical statistics, Banach Cent. Publ., 6, 1980, 121–132 | MR | Zbl

[33] I. I. Gikhman, A. V. Skorokhod, Teoriya sluchainykh protsessov, t. I, izd-vo “Nauka”, Moskva, 1971 | MR