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Minimising the time to a majority decision

Le : 01/02/2010 11h00
Par : Peter WINDRIDGE (Warwick)
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Résumé : We consider a stochastic control problem in which the controller seeks to minimise the time taken for a Brownian populace to reach a majority decision. More specifically, suppose that we are given three independent Brownian motions on the unit interval with absorption at the endpoints. At each instant, one must choose which of the three Brownian motions to run, the objective being to minimise the time taken for at least two to be absorbed at the same value. We propose a candidate optimal strategy and then, using a heuristic argument show it attains $\sup_c \mathbb{E}_x(e^{-r au_c})$ (where the supremum is taken over all controls, $au_c$ is the corresponding decision time and > 0\$). We conclude by deducing \emph{stochastic minimality} of the decision time for our strategy.