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Limit at zero of the brownian first-passage density

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  • 45 Integral Equations
  • 60 Probability Theory And Stochastic Processes
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  • Mathematics

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.

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