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Seminar on Stochastic Processes, 1982 pp 219–227Cite as

Remarks on the Convex Minorant of Brownian Motion

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Part of the book series: Progress in Probability and Statistics ((PRPR,volume 5))

Abstract

Recently Groeneboom [1] studied the concave majorant process of a Brownian motion (Bt, t ≤ O). The purpose of this note is to take a fresh look at some of Groeneboom’s results in the context of path decompositions of Williams [7], and to give a simple new description of this concave majorant process.

Research supported by NSF Grant No. MCS 82-02552.

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References

  1. P. Groeneboom. The concave majorant of Brownian motion. Technical Report No. 6, Dept. Statistics, Univ. of Washington, Seattle. To appear in Ann. Probab.

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  2. B. Maisonneuve. Exit systems. Ann. Probab. 3 (1975), 399–411.

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  3. J.W. Pitman and M. Yor. Bessel processes and infinitely divisible laws. Stochastic Integrals, ed. D. Williams, 285–292. Lecture Notes in Math. 851. Springer-Verlag, Berlin, 1981.

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  4. L.C.G. Rogers and J.W. Pitman. Markov functions. Ann. Probab. 9 (1981), 573–582.

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  5. S. Watanabe. On time inversion of one-dimensional diffusion processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31 (1975), 115–124.

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  6. D. Williams. Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 (1970), 871–873.

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  7. D. Williams. Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. London Math. Soc., Ser. 3, 28 (1974), 738–768.

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Author information

Authors and Affiliations

  1. Department of Statistics, Univ. of California at Berkeley, Berkeley, CA, 94720, USA

    J. W. Pitman

Authors
  1. J. W. Pitman
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Editor information

Editors and Affiliations

  1. Technological Institute, Northwestern University, Evanston, Illinois, 60201, USA

    E. Çinlar

  2. Department of Mathematics, Stanford University, Stanford, California, 94305, USA

    K. L. Chung

  3. Department of Mathematics, University of California — San Diego, La Jolla, California, 92093, USA

    R. K. Getoor

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© 1983 Birkhäuser, Boston, Inc.

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Pitman, J.W. (1983). Remarks on the Convex Minorant of Brownian Motion. In: Çinlar, E., Chung, K.L., Getoor, R.K. (eds) Seminar on Stochastic Processes, 1982. Progress in Probability and Statistics, vol 5. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-0540-8_11

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  • DOI: https://doi.org/10.1007/978-1-4684-0540-8_11

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-3131-4

  • Online ISBN: 978-1-4684-0540-8

  • eBook Packages: Springer Book Archive

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