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

When the $\mathit{\text{martingale}}\mathit{\text{representation}}\mathit{\text{property}}$ holds, we call any local martingale which realizes the representation a $\mathit{\text{representation}}\mathit{\text{process}}$. There are two properties of the $\mathit{\text{representation}}\mathit{\text{process}}$ which can greatly facilitate the computations under the \textit{martingale representation property}. Actually, on the one hand, the $\mathit{\text{representation}}\mathit{\text{process}}$ is not unique and there always exists a $\mathit{\text{representation}}\mathit{\text{process}}$ which is locally bounded and has pathwisely orthogonal components outside of a predictable thin set. On the other hand, the jump measure of a $\mathit{\text{representation}}\mathit{\text{process}}$ 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 $\mathit{\text{full}}\mathit{\text{viability}}$ of an expansion of market information flow implies the $\mathit{\text{drift}}\mathit{\text{multiplier}}\mathit{\text{assumption}}$.