Limit at zero of the brownian first-passage density

Peskir, Goran (2002) Limit at zero of the brownian first-passage density. Probability Theory and Related Fields, 124 (1). pp. 100-111. ISSN 1432-2064

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Abstract

Let (Bt)t S 0) be a standard Brownian motion started at zero, let g : Â_+ M Â be an upper function for B satisfying g(0)=0, and let $$\tau = \inf \, \{ \; t > 0 \; \vert \; B_t \ge g(t) \, \}$$ be the first-passage time of B over g. Assume that g is C1 on d0,X¢, increasing (locally at zero), and concave (locally at zero). Then the following identities hold for the density function f of F: $$ f(0+) = \lim_{t \downarrow 0} {1 \over 2} {{g(t)} \over t^{3/2}} \varphi\bigg({{g(t)} \over \sqrt{t}}\bigg) = lim_{t \downarrow 0} {{g'(t)} \over \sqrt{t}} \varphi\bigg({{g(t)} \over \sqrt{t}}\bigg)$$ in the sense that if the second and third limit exist so does the first one and the equalities are valid (here $\varphi(x)=(1/\sqrt{2 \pi }) e^{-x^2/2}$ is the standard normal density). These limits can take any value in [0,X]. The method of proof relies upon the strong Markov property of B and makes use of real analysis.

Item Type:	Article
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 45 Integral equations MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
Depositing User:	Ms Lucy van Russelt
Date Deposited:	21 Jul 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/405

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