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International Journal of Stochastic Analysis
Volume 2012, Article ID 687376, 14 pages
http://dx.doi.org/10.1155/2012/687376
Research Article

Consistent Price Systems in Multiasset Markets

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA

Received 27 May 2012; Accepted 9 July 2012

Academic Editor: Lukasz Stettner

Copyright © 2012 Florian Maris and Hasanjan Sayit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑋 𝑡 be any d-dimensional continuous process that takes values in an open connected domain 𝒪 in 𝑑 . In this paper, we give equivalent formulations of the conditional full support (CFS) property of 𝑋 𝑡 in 𝒪 . We use them to show that the CFS property of X in 𝒪 implies the existence of a martingale M under an equivalent probability measure such that M lies in the 𝜖 > 0 neighborhood of 𝑋 𝑡 for any given 𝜖 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.