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We provide, in a general setting, explicit solutions for optimal stoppingproblems that involve a diffusion process and its running maximum. Besides, anew feature includes absorbing boundaries that vary with the value of therunning maximum. The existence of the absorbing boundary of this type makes theproblem harder but more practical and flexible. Our approach is to use theexcursion theory for Levy processes. Since general diffusions are, inparticular, not of independent increments, we use an appropriate measure changeto make the process have that property. Then we rewrite the originaltwo-dimensional problem as an infinite number of one-dimensional ones andcomplete the solution. We show general solution methods with explicit valuefunctions and corresponding optimal strategies, illustrating them by someexamples.

Explicit Solutions for Optimal Stopping of Maximum Process with Absorbing Boundary that Varies with It