The hazard rate tangent approximation for boundary hitting times. (English) Zbl 0836.60087

The approximation of the distribution of the hitting time \(\tau_f = \inf_{t > 0} \{t : X_t \geq f(t)\}\) of a diffusion process \(X_t\) is considered. When \(X_t\) is the standard Brownian motion \(B_t\), the tangent approximation provides an inaccurate density approximation of \(\tau_f\) by using the Bachelier-Levy formula: \(p^{a,b} (t) = {a \over t^{3/2}} \varphi ({a + bt \over t^{1/2}})\), where \(\varphi\) is the standard Brownian density, \(a = f(t) - tf'(t)\), \(b = f'(t)\). Authors introduce the hazard rate tangent approximation, which considerably improves on the tangent approximation when \(f\) is convex or concave. Numerical examples are also given.


60J65 Brownian motion
60J50 Boundary theory for Markov processes
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