Examples of optimal prediction in the infinite horizon case

Timing of financial decisions, especially in a volatile market such as the one we are in now, is crucial to maintaining and growing wealth. Without premonition or inside information, buyers and sellers of financial assets may experience a form of remorse for selling too late or buying too early. A valuable tool in reducing such regret is an algorithm that tells the asset holder when to sell “optimally”. In the case of a Brownian-valued asset, Graversen et al. (2000) proposed the strategy of Optimal Prediction  , where the Brownian motion is stopped as close as possible, in the mean-square sense, to its ultimate maximum over the entire term [0,T][0,T]. Any candidate for the optimal stopping time must be adapted to the filtration of the underlying asset, since no inside information is to be assumed. Later work, nicely summarized in the book of Peskir and Shiryaev (2006), has extended this field to include different non-adapted functionals and different measures of “closeness”. In this article, we seek to extend the field of optimal prediction to the perpetual, or infinite horizon, case. Some examples related to the ultimate risk associated with holding a toxic liability and the ultimately best time to sell a stock are presented, and their closed form solutions are computed.