Absolute continuity of measures corresponding to the Ito processes and processes of the diffusion type

  • R. S. Liptser
  • A. N. Shiryayev
Part of the Applications of Mathematics book series (SMAP, volume 5)


Let (0, F, P) be a complete probability space, let F = (F t), t ≥ 0, be a nondecreasing family of sub-σ-algebras, and let W = (W t, F t), t ≥ 0, be a Wiener process.


Gaussian Process Wiener Process Diffusion Type Absolute Continuity Complete Probability Space 
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Notes and references

  1. [22]
    Wolfowitz J., On sequential binomial estimation. AMS 17 (1946), 489–493.MathSciNetzbMATHGoogle Scholar
  2. [35]
    Gikhman I. I., Skorokhod A. V., On densities of probability measures on function spaces. UMN 21 (1966), 83–152.zbMATHGoogle Scholar
  3. [46]
    Doob J. L., Probability Processes. Russian transi., IL. Moscow, 1956.Google Scholar
  4. [52]
    Yershov M. P., On representations of Ito processes. Teoria Verojatn. i Primenen. XVII, 1 (1972), 167–172.Google Scholar
  5. [53]
    Yershov M. P., On absolute continuity of measures corresponding to diffusion type processes. Teoria Verojatn. i Primenen. XVII, 1 (1972), 173–178.Google Scholar
  6. [64]
    Kadota T. T., Nonsingular Detection and Likelihood Ratio for Random Signals in White Gaussian Noise. IEEE Trans. Inform. Theory IT-16 (1970), 291–298.Google Scholar
  7. [66]
    Kadota T. T., Shepp L. A., Conditions for the absolute continuity between a certain pair of probability measures. Z. Wahrscheinlickkeitstheorie verw. Gebiete 16, 3 (1970), 250–260.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [69]
    Kailath T., The structure of Radon-Nykodym derivatives with respect to Wiener and related measures. AMS 42 (1971), 1054–1067.MathSciNetzbMATHGoogle Scholar
  9. [67]
    Kailath T., An innovations approach to least-squares estimation, Parts I, II. IEEE Trans. Automatic Control AC-13 (1968), 646–660.Google Scholar
  10. [74]
    Kallianpur G., Striebel C., Estimation of stochastic systems: Arbitrary system process with additive white noise observation errors. AMS 39 (1968), 785–801.MathSciNetzbMATHGoogle Scholar
  11. [80]
    Cameron R. H., Martin W. T., Transformation of Wiener integrals under a general class of linear transformations. Trans. Amer. Math. Soc. 58 (1945), 184–219.MathSciNetzbMATHGoogle Scholar
  12. [81]
    Cameron R. H., Martin W. T., Transformation of Wiener integrals by nonlinear transformation. Trans. Amer. Math. Soc. 66 (1949), 253–283.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [90]
    Cramer G., Mathematical Statistics Methods. Russian transi., IL, Moscow, 1948.Google Scholar
  14. [103]
    Leonov V. P., Shiryayev A. N., Methods for calculating semi-invariants. Teoria Verojatn. i Primenen. IV, 2 (1959), 342–355.Google Scholar
  15. [111]
    Liptser R. S., Shiryayev A. N., Nonlinear filtering of diffusion type Markov processes. Trudy matem. in-ta im. V. A. Steklova AN SSSR 104 (1968), 135–180.Google Scholar
  16. [118]
    Liptser R. S., Shiryayev A. N., On absolute continuity of measures corresponding to diffusion type processes with respect to a Wiener measure. Izv. AN SSSR, ser. matem. 36, 4 (1972), 874–889.Google Scholar
  17. [134]
    Prokhorov Yu. V., Convergence of random processes and limit theorems of probability theory. Teoria Verojatn. i Primenen. I, 2 (1956), 177–238.Google Scholar
  18. [156]
    Fujisaki M., Kallianpur G., Kunita H., Stochastic differential equations for the nonlinear filtering problem. Osaka J. Math. 9, 1 (1972), 19–40.MathSciNetzbMATHGoogle Scholar
  19. [158]
    Hitsuda M., Representation of Gaussian processes equivalent to Wiener processes. Osaka J. Math. 5 (1968), 299–312.MathSciNetzbMATHGoogle Scholar
  20. [164]
    Shiryayev A. N., Problems of spectral theory of higher moments I. Teoria Verojatn. i Primenen. V, 3 (1960), 293–313.Google Scholar
  21. [166]
    Shiryayev A. N., Stochastic equations of nonlinear filtering of jump Markov processes. Problemy peredachi informatsii. II, 3 (1966), 3–22.Google Scholar

Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • R. S. Liptser
    • 1
  • A. N. Shiryayev
    • 2
  1. 1.Institute for Problems of Control TheoryMoscowUSSR
  2. 2.Institute of Control SciencesMoscowUSSR

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