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Exact asymptotics for boundary crossing probabilities of Brownian motion with piecewise linear trend. (English) Zbl 1110.60076

Summary: Let \(B\) be a standard Brownian motion and let \(b_\gamma\) be a piecewise linear continuous boundary function. We obtain an exact asymptotic expansion of \({\mathbf P}\{B(t)<b_\gamma(t)\), \(\forall t \in[0,1]\}\) provided that the boundary function satisfies \(\lim_{\gamma \to\infty}b_\gamma(t^*)=-\infty\) for some \(t^*\in(0,1]\).

MSC:

60J65 Brownian motion
60F10 Large deviations
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