The Annals of Probability

Decreasing sequences of $\sigma$-fields and a measure change for Brownian motion

Lester Dubins, Jacob Feldman, Meir Smorodinsky, and Boris Tsirelson

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Abstract

Let $(F_t)_{t \geq 0}$ be the filtration of a Brownian motion $(B(t))_{t \geq 0}on $(\Omega,F,P)$. An example is given of a measure $Q \sim P$ (in the sense of absolute continuity) for which $(F_t)_{t \geq 0}$ is not the filtration of any Brownian motion on $(\Omega,F,Q)$. This settles a 15-year-old question.

Article information

Source
Ann. Probab., Volume 24, Number 2 (1996), 882-904.

Dates
First available in Project Euclid: 11 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1039639367

Digital Object Identifier
doi:10.1214/aop/1039639367

Mathematical Reviews number (MathSciNet)
MR1404533

Zentralblatt MATH identifier
0870.60078

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 60G07: General theory of processes 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Brownian filtration equivalent measure decreasing sequence of measurable partitions

Citation

Dubins, Lester; Feldman, Jacob; Smorodinsky, Meir; Tsirelson, Boris. Decreasing sequences of $\sigma$-fields and a measure change for Brownian motion. Ann. Probab. 24 (1996), no. 2, 882--904. doi:10.1214/aop/1039639367. https://projecteuclid.org/euclid.aop/1039639367


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References

  • DUDLEY, R. M. 1989. Real Analy sis and Probability. Wadsworth and Brooks Cole, Pacific Grove, CA. Z.
  • GETOOR, R. and SHARPE, M. 1972. Conformal martingales. Invent. Math. 16 271 308. Z.
  • IKEDA, N. and WATANABE, S. 1989. Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland, Amsterdam Kodansha, Toky o. Z. ()
  • KAKUTANI, S. 1948. On equivalence of infinite product measures. Ann. of Math. 2 49 214 224. Z.
  • KANTOROVICH, L. V. and RUBINSTEIN, G. S. 1958. On space of completely additive functions. Z. Vestnik Leningrad Univ. 7 52 59. In Russian. Z.
  • PROTTER, P. 1990. Stochastic Integration and Differential Equations. Springer, Berlin. Z.
  • REVUZ, D. and YOR, M. 1991. Continuous Martingales and Brownian Motion. Springer, Berlin. Z.
  • ROGERS, L. C. G. and WILLIAMS, D. 1987. Diffusions, Markov Processes, and Martingales 2. Ito Calculus. Wiley, New York. Z. Z.
  • ROKHLIN, V. A. 1949. Main notions of measure theory. Mat. Sb. 67 107 150. In Russian. Z.
  • SKOROKHOD, A. V. 1986. Random processes in infinite dimensional spaces. Proc. Internat. Cong. Z. Z. Math. A. M. Gleason, ed. 163 171. Amer. Math. Soc., Providence, RI. In Russian. Z.
  • SMORODINSKY, M. 1995. An example of a nonstandard inverse filtration. Preprint.
  • DUBINS, FELDMAN, SMORODINSKY AND TSIRELSON 904
  • STROOCK, D. W. and YOR, M. 1980. On extremal solutions of martingale problems. Ann. Sci. ´ () Ecole Norm. Sup. 4 13 95 164. Z.
  • TSIRELSON, B. S. 1975. An example of a stochastic differential equation having no strong solution. Theory Probab. Appl. 20 416 418. Z.
  • VERSHIK, A. M. 1968. A theorem on lacunary isomorphisms. Functional Anal. Appl. 2 200 203. Z.
  • VERSHIK, A. M. 1970. Decreasing sequences of measurable partitions, and their applications. Soviet Math. Dokl. 11 1007 1011. Z.
  • VERSHIK, A. M. 1971. A continuum of pairwise nonisomorphic dy adic sequences. Functional Anal. Appl. 5 16 18. Z. Z
  • VERSHIK, A. M. 1973. Approximation in measure theory. Dissertation, Leningrad Univ. In. Russian. Z.
  • VERSHIK, A. M. 1994. The theory of decreasing sequences of measurable partitions. St. Petersburg Math. J. 6 705 761.
  • BERKELEY, CALIFORNIA 94720 TEL AVIV 69978 E-MAIL: dubins@stat.berkeley.edu ISRAEL feldman@math.berkeley.edu