One often encounters options involving not only the stock price, but also its running maximum. We provide, in a fairly general setting, explicit solutions for
optimal stopping problems concerned with a diffusion process and its running maxi-mum. Our approach is to use excursion theory for Markov processes and rewrite the original two-dimensional problem as an infinite number of one-dimensional ones. Our method is rather direct without presupposing the existence of an optimal thresh- old or imposing a smooth-fit condition. We present a systematic solution method by illustrating it through classical and new examples.

EB00000003270K

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