Martingale representation processes and applications in the market viability under information flow expansion

Abstract

Abstract. When the martingale representation property holds, we call any local martingale which realizes the representation a representation process. There are two properties of the representation process which can greatly facilitate the computations under the martingale representation property. On the one hand, the representation process is not unique and there always exists a representation process which is locally bounded and has pathwise orthogonal components outside of a predictable thin set. On the other hand, the jump measure of a representation process satisfies the finite predictable constraint, which implies the martingale projection property. In this paper, we give a detailed account of these properties. As application, we will prove that, under the martingale representation property, the full viability of an expansion of market information flow implies the drift multiplier assumption.

Drift operator in a viable expansion of information flow

“…Let us check that the conditions in [45,Theorem 3.8] is satisfied for the jump measure of W ′′ . Recall…”

“…[45], the full viability implies that, for any (P, F) locally bounded local martingale M, M satisfies the raw structure condition in (P, G) on [0, T ]. 2. there exist N = (N 1 , .…”

“…According to [45], under the martingale representation property, the full viability on [0, T ] implies the drift multiplier assumption. The aim of this paper is to refine the above result to have an exact relationship between the drift multiplier assumption and the full viability of the expanded information flow.…”

“…Suppose the martingale representation property in (P, F). Reconstituting the original representation process if necessary, we can find a particular representation process in the form W = (W ′ , W ′′ , W ′′′ ) (in juxtaposition of three (multi-dimensional) processes), where W ′ , W ′′ denote respectively the processes defined in [45,Formulas (4) and (5) …”

“…In fact, when a process W has the martingale representation property, the jump ∆W of this process can only take a finite number of "predictable" values. We refer to [45] for a detailed account, where it is called the finite predictable constraint condition (which has a closed link with the notion of multiplicity introduced in [6]). …”