On characterizing the set of martingale measures in discrete time

Abstract

We state necessary and sufficient conditions on a set of probability measures to be the set of martingale measures for a vector valued, bounded and adapted process. In the absence of the maximality condition, we prove the existence of the smallest set of martingale measures. We apply such results to the finite sample space case.

On representations of the set of supermartingale measures and applications in discrete time

“…Proof For t ∈ I * we denote by Q t for t ∈ I * , the set of probability measures Q ∈ P, defined on ( , t+1 . It has been proved in the proofs of Theorems 4.1 and 4.2 that the assumption MRP(S t , S t+1 ) is satisfied on the one-period model {t, t + 1} with t S := S t+1 − S t and that Q t is optionally m-stable with respect to t S. We apply then Proposition 3.3 in [2] and obtain that…”

“…The set Q was first introduced by Berkaoui in [2] and defined properly in [3]. We refer to Theorem 2.1 in [3] for more details on this set.…”

“…In [2] it was shown that the m-stability assumption on Q is a corner stone in generalizing a number of results from one-period case to multi-period case. We recall in particular that for an m-stable set Q, we get the following: (i) Suppose Q = {Q ∈ P : E Q (X ) = 0} =: M(0, X ) for a vector-valued random variable X = (X 1 , .…”