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. We present necessary and sufficient conditions for the solution, establishing existence and uniqueness. We show that the free boundary is star-shaped, and present  asymptotic characterization of the value function and the free boundary. We also show properties of how the distance between the free boundary and the diagonal varies with the number of alternatives  .
We consider the problem of a decision-maker searching for information on multiple alternatives when information is learned on all alternatives simultaneously. The decision-maker has 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. We present necessary and sufficient conditions for the solution, establishing existence and uniqueness. We show that the optimal boundary where search is stopped (free boundary) is star-shaped, and present an asymptotic characterization of the value function and the free boundary. We show properties of how the distance between the free boundary and the diagonal varies with the number of alternatives , and how the free boundary under parallel search relates to the one under sequential search, with and without economies of scale on the search costs .
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an optimal stopping
the
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d-dimensional Brownian motion, where the payoff at stopping is the maximum component of the Brownian motion, and there is
decision-maker searching for information on multiple alternatives when information is learned on all alternatives simultaneously. The decision-maker has
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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.
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.
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free boundary
optimal boundary where search is stopped (free boundary)
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an
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also
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, and how the free boundary under parallel search relates to the one under sequential search, with and without economies of scale on the search costs