A Martingale Representation Result and an Application to Incomplete Financial Markets

Abstract

We establish necessary and sufficient conditions for an H 1 ‐martingale to be representable with respect to a collection, of local martingales. M H 1 ( P ) is representable if and only if M is a local martingale under all p.m.'s Q which are “uniformly equivalent” to P and which make all the elements of local martingales (Theorem 1.1). We then give necessary and sufficient conditions which are easier to verify, and only involve expectations (Theorem 1.2). We go on to apply these results to the problem of pricing claims in an incomplete financial market‐establishing two conjectures of Harrison and Pliska(1981).