Summary
Let p(t) be the density of the first-exit time of a Brownian motion over the one-sided moving boundary given by x=f(t). We derive the following formal expansion for p:
Here λ(t)=f(t)−f′(t), ϕ is the standard normal density, m n is the Hermite function of order (−n), and the coefficients c n are functions of the derivatives of f at t. We give bounds for the error incurred by approximating p by the first n terms of the series, and examples in which the series provides an asymptotic expansion for p.
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Work supported by the Deutsche Forschungsgemeinschaft at the Sonderforschungsbereich 123, Universität Heidelberg
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Ferebee, B. An asymptotic expansion for one-sided Brownian exit densities. Z. Wahrscheinlichkeitstheorie verw Gebiete 63, 1–15 (1983). https://doi.org/10.1007/BF00534172
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DOI: https://doi.org/10.1007/BF00534172