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Limit at zero of the Brownian first-passage density

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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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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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Keywords

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