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Discounted optimal stopping for maxima in diffusion models with finite horizon. (English) Zbl 1127.60037
Summary: We present a solution to some discounted optimal stopping problem for the maximum of a geometric Brownian motion on a finite time interval. The method of proof is based on reducing the initial optimal stopping problem with the continuation region determined by an increasing continuous boundary surface to a parabolic free-boundary problem. Using the change-of-variable formula with local time on surfaces we show that the optimal boundary can be characterized as a unique solution of a nonlinear integral equation. The result can be interpreted as pricing American fixed-strike lookback options in a diffusion model with finite time horizon.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
35R35 Free boundary problems for PDEs
45G10 Other nonlinear integral equations
60J60 Diffusion processes
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