标题： 有限变差的$G$-鞅的性质及其在$G$-Sobolev空间中的应用 显示英文标题

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

摘要： 众所周知，形式为$\int_0^t\eta_sd\langle B\rangle_s-\int_0^t2G(\eta_s)ds$，$\eta\in M^1_G(0,T)$的过程是一个非增$G$-鞅。 在本文中，我们将证明一个非增的$G$-鞅不能是$\int_0^t\eta_sds$或$\int_0^t\gamma_sd\langle B\rangle_s$,$\eta, \gamma \in M^1_G(0,T)$的形式，这表明广义的$G$-伊藤过程的分解是唯一的： 对于$\zeta\in H^1_G(0,T)$,$\eta\in M^1_G(0,T)$和非增的$G$-鞅$K, L$，如果\[\int_0^t\zeta_s dB_s+\int_0^t\eta_sds+K_t=L_t,\ t\in[0,T],\]，那么我们有$\eta\equiv0$,$\zeta\equiv0$和$K_t=L_t$。 作为应用，我们给出了Peng和Song（2015）引入的$G$-Sobolev空间的一个特征。 显示英文摘要

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