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The intrinsic local time sheet of Brownian motion
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  • Published: September 1991

The intrinsic local time sheet of Brownian motion

  • L. C. G. Rogers1 &
  • J. B. Walsh2 

Probability Theory and Related Fields volume 88, pages 363–379 (1991)Cite this article

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  • 10 Citations

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Summary

McGill showed that the intrinsic local time process\(\tilde L\)(t, x), t ≧ 0, x ∈ ℝ, of one-dimensional Brownian motion is, for fixedt>0, a supermartingale in the space variable, and derived an expression for its Doob-Meyer decomposition. This expression referred to the derivative of some process which was not obviously differentiable. In this paper, we provide an independent proof of the result, by analysing the local time of Brownian motion on a family of decreasing curves. The ideas involved are best understood in terms of stochastic area integrals with respect to the Brownian local time sheet, and we develop this approach in a companion paper. However, the result mentioned above admits a direct proof, which we give here; one is inevitably drawn to look at the local time process of a Dirichlet process which is not a semimartingale.

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References

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Authors and Affiliations

  1. Statistical Laboratory, 16 Mill Lane, CB2 1SB, Cambridge, UK

    L. C. G. Rogers

  2. Department of Mathematics, University of British Columbia, V6T 1Y4, Vancouver, Canada

    J. B. Walsh

Authors
  1. L. C. G. Rogers
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  2. J. B. Walsh
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Rogers, L.C.G., Walsh, J.B. The intrinsic local time sheet of Brownian motion. Probab. Th. Rel. Fields 88, 363–379 (1991). https://doi.org/10.1007/BF01418866

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  • Received: 10 March 1990

  • Revised: 02 November 1990

  • Issue Date: September 1991

  • DOI: https://doi.org/10.1007/BF01418866

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Keywords

  • Time Process
  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Local Time
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