Uniqueness of the representation for $G$-martingales with finite variation

Abstract

Letting $\{\delta_n\}$ be a refining sequence of Rademacher functions on the interval $[0,T]$, we introduce a functional on processes in the $G$-expectation space by $d(K)=\limsup_n\hat{E}[\int_0^T\delta_n(s)dK_s].$ We prove that $d(K)>0$ if $K_t=\int_0^t\eta_sd\langle B\rangle_s$ with nontrivial $\eta\in M^1_G(0,T)$ and that $d(K)=0$ if $K_t=\int_0^t\eta_sds$ with $\eta\in M^1_G(0,T)$. This implies the uniqueness of the representation for $G$-martingales with finite variation, which is the main purpose of this article.