| Summary: | This paper studies stopping problems of the form $V=\inf_{0 \leq \tau \leq T} \mathbb{E}[U(\frac{\max_{0\le s \le T} Z_s }{Z_\tau})]$ for strictly concave or convex utility functions U in a family of increasing functions satisfying certain conditions, where Z is a geometric Brownian motion and T is the time of the nth jump of a Poisson process independent of Z. We obtain some properties of $V$ and offer solutions for the optimal strategies to follow. This provides us with a technique to build numerical approximations of stopping boundaries for the fixed terminal time optimal stopping problem presented in [J. Du Toit and G. Peskir, Ann. Appl. Probab., 19 (2009), pp. 983--1014].
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