Computing the Expectation of the Azéma-Yor Stopping Times
نویسنده
چکیده
Given the maximum process (St) = (max 0≤r≤t Xr) associated with a diffusion ((Xt),Px), and a continuous function g satisfying g(s) < s, we show how to compute the expectation of the Azéma-Yor stopping time τg = inf{ t > 0 : Xt ≤ g(St) } as a function of x. The method of proof is based upon verifying that the expectation solves a differential equation with two boundary conditions. The third ‘missing’ condition is formulated in the form of a minimality principle which states that the expectation is the minimal non-negative solution to this system. It enables us to express this solution in a closed form. The result is applied in the case when (Xt) is a Bessel process and g is a linear function.
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