COMPUTING THE EXPECTATION OF THE AZEMA-YOR STOPPING-TIMES

Given the maximum process (S-t) = (max(0 less than or equal to r less than or equal to t) X-r) associated with a diffusion ((X-t), P-x), and a continuous function g satisfying g(s) < s, we show how to compute t he expectation of the Azema-Yor stopping time tau(g) = inf{t > 0 \ X-t less than or equal to g(S-t)} as a function of x. The method of proof is based upon verifying that the expectation solves a differential eq uation 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 e nables us to express this solution in a closed form. The result is app lied in the case when (X-t) is a Bessel process and g is a linear func tion. (C) Elsevier, Paris.