## 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.

### MSC:

 60J65 Brownian motion 60J50 Boundary theory for Markov processes
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