Consistent Price Systems in Multiasset Markets

Let 𠑋 𠑡 be any d -dimensional continuous process that takes values in an open connected domain 𠒪 in ℠𠑑 . In this paper, we give equivalent formulations of the conditional full support (CFS) property of 𠑋 𠑡 in 𠒪 . We use them to show that the CFS property of X in 𠒪 implies the existence of a martingale M under an equivalent probability measure such that M lies in the 𠜖 > 0 neighborhood of 𠑋 𠑡 for any given 𠜖 under the supremum norm. The existence of such martingales, which are called consistent price systems (CPSs), has relevance with absence of arbitrage and hedging problems in markets with proportional transaction costs as discussed in the recent paper by Guasoni et al. (2008), where the CFS property is introduced and shown sufficient for CPSs for processes with certain state space. The current paper extends the results in the work of Guasoni et al. (2008), to processes with more general state space.