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1998•Journal of Applied ProbabilityRequires access

On a formula of Takács for Brownian motion with drift

R. A. Doney, Marc Yor

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Abstract

A recent result of Takács (1995) gives explicitly the density of the time spent beforetabove a levelx≠ 0 by Brownian motion with drift. Takács' proof is by means of random walk approximations to Brownian motion, but in this paper we give two different proofs of this result by considerations involving only Brownian motion. We also give a reformulation of Takács' result which involves Brownian meanders, and an extension of Denisov's representation of Brownian motion in terms of two independent Brownian meanders.

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What this paper is about

A recent result of Takács (1995) gives explicitly the density of the time spent beforetabove a levelx≠ 0 by Brownian motion with drift. Takács' proof is by means of random walk approximations to Brownian motion, but in this paper we give two different proofs of this result by considerations involving only Brownian motion. We also give a reformulation of Takács' result which involves Brownian meanders, and an extension of Denisov's representation of Brownian motion in terms of two independent Brownian meanders.

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Available abstract

A recent result of Takács (1995) gives explicitly the density of the time spent beforetabove a levelx≠ 0 by Brownian motion with drift. Takács' proof is by means of random walk approximations to Brownian motion, but in this paper we give two different proofs of this result by considerations involving only Brownian motion. We also give a reformulation of Takács' result which involves Brownian meanders, and an extension of Denisov's representation of Brownian motion in terms of two independent Brownian meanders.

Key concepts: Mathematics, Brownian motion, Martingale representation theorem, Reflected Brownian motion, Brownian excursion, Geometric Brownian motion, Diffusion process, Heavy traffic approximation

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