Cooperative Stopping Rules in Multivariate Problems

Abstract

Consider a sequence of random vectors Y(1),…, Y(n) in ℛ d , with a known joint distribution, finite expectations, adapted to a filtration ℱ. For a given monotone function h : ℛ d → ℛ, a randomized stopping rule t ⋆ such that sup t h(E Y(t)) = h(E Y(t ⋆)) is desired. A full solution to this problem is given. In particular we show that there exists a vector of constants a = (a 1,…, a d ) such that the optimizing t ⋆ is optimal also for the one-dimensional optimal stopping problem, where the sequence of rewards is Z(1),…, Z(d) with Z(j) = . With symmetry assumptions the a vector will be (1,…, 1). The results are extended also to the infinite horizon case and are applied to obtain an explicit solution to the multivariate “house-selling problem.”