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We provide, in a general setting, explicit solutions for optimal stoppingproblems that involve diffusion process and its running maximum. Our approachis to use the excursion theory for Levy processes. Since general diffusionsare, in particular, not of independent increments, we use an appropriatemeasure change to make the process have that property. Then we rewrite theoriginal two-dimensional problem as an infinite number of one-dimensional onesand complete the solution. We show general solution methods with explicit valuefunctions and corresponding optimal strategies, illustrating them by someexamples.

Explicit Solutions for Optimal Stopping of Linear Diffusion and its Maximum