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

On the degree of incompleteness of an incomplete financial market

“…Conversely suppose that A = A(X) for X ∈ C n , then thanks to Lemma 4.3 in [1], we get A = A(X 1 ) + . .…”

“…So in order to define the dimension of an element A ∈ b, we should define first the components that constitute A and define the dimension of A as the minimum number of these components needed to generate the set A. In [1], the notion of a section of A was introduced and it was shown that only a building block of A is a section of A. So on the basis of this concept we define a component as a set A ∈ b such that any subset B ∈ b in A is a section of A and define the dimension of a general set A ∈ b as the minimum number of components A 1 , .…”

“…Review. We recall the notion of a section of an acceptance set introduced in [1] and review some of its properties.…”

“…Definition 1.1. (Berkaoui [1])Let F ∈ P and A ∈ a with the associated set Q of test probabilities. We define the F -section of A as follows:…”

“…It has been proved in [1] that 1 F • A ∈ a, we denote by 1 F • Q, the associated set of test probabilities. We shall say that B ∈ a is a section of A if it is an F -section of A for some…”