Before revision: We consider an optimal stopping problem of a d-dimensional Brownian motion, where the payoff at stopping is the maximum component of the Brownian motion, and there is a running cost before stopping. Applications include choosing one among several alternatives while learning simultaneously about all the alternatives (parallel search), and exercising an option based on several assets.
After revision: We consider an optimal stopping problem of a d-dimensional Brownian motion, where the payoff at stopping is the maximum component of the Brownian motion, and there is a running cost of searching for information, and has to decide when to stop searching for information and choose one alternative. The expected payoff of each alternative evolves as a diffusion process when information is being learned.