On the existence of equivalent τ-measures in finite discrete time

Suppose that (X(n)) is a finite adapted sequence of d-dimensional random variables defined on some filtered probability space (Ω, F, (Fn), P). We obtain conditions which are necessary and sufficient for the existence of a probability measure Q equivalent to P (which we call an equivalent τ-measure) such that each of the d component sequences of (X(n)) has a prescribed martingale property w.r.t. Q (i.e., it is either a Q-martingale, a Q-sub- or a Q-supermartingale). This extends a version of the Fundamental Theorem of Asset Pricing due to Dalang et al. (1990).