Paper detail

Properties of $G$-martingales with finite variation and the application to $G$-Sobolev spaces

As is known, a process of form $\int_0^tη_sd\langle B\rangle_s-\int_0^t2G(η_s)ds$, $η\in M^1_G(0,T)$, is a non-increasing $G$-martingale. In this paper, we shall show that a non-increasing $G$-martingale could not be form of $\int_0^tη_sds$ or $\int_0^tγ_sd\langle B\rangle_s$, $η, γ\in M^1_G(0,T)$, which implies that the decomposition for generalized $G$-Itô processes is unique: For $ζ\in H^1_G(0,T)$, $η\in M^1_G(0,T)$ and non-increasing $G$-martingales $K, L$, if \[\int_0^tζ_s dB_s+\int_0^tη_sds+K_t=L_t,\ t\in[0,T],\] then we have $η\equiv0$, $ζ\equiv0$ and $K_t=L_t$. As an application, we give a characterization to the $G$-Sobolev spaces introduced in Peng and Song (2015).