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

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