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Limit at zero of the Brownian first-passage density
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  • Published: September 2002

Limit at zero of the Brownian first-passage density

  • Goran Peskir1 

Probability Theory and Related Fields volume 124, pages 100–111 (2002)Cite this article

Abstract.

 Let (B t ) t ≥ 0) be a standard Brownian motion started at zero, let g : ℝ_+ →ℝ be an upper function for B satisfying g(0)=0, and let

be the first-passage time of B over g. Assume that g is C 1 on <0,∞>, increasing (locally at zero), and concave (locally at zero). Then the following identities hold for the density function f of τ:

in the sense that if the second and third limit exist so does the first one and the equalities are valid (here is the standard normal density). These limits can take any value in [0,∞]. The method of proof relies upon the strong Markov property of B and makes use of real analysis.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, 8000 Aarhus, Denmark. e-mail: goran@imf.au.dk; home.imf.au.dk/goran, , , , , , DK

    Goran Peskir

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  1. Goran Peskir
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Additional information

Received: 30 August 2001 / Revised version: 25 February 2002 / Published online: 22 August 2002

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Peskir, G. Limit at zero of the Brownian first-passage density. Probab Theory Relat Fields 124, 100–111 (2002). https://doi.org/10.1007/s004400200208

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  • Issue Date: September 2002

  • DOI: https://doi.org/10.1007/s004400200208

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Keywords

  • Density Function
  • Brownian Motion
  • Normal Density
  • Markov Property
  • Real Analysis
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