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

Abstract

As is known, a process of form ${\int }_0^t{\eta }_{s}d⟨B{⟩}_s-{\int }_0^{t}2G({\eta }_s)ds$, $\eta \in M_G^1(0,T)$, is a non-increasing $G$-martingale. In this paper, we shall show that a non-increasing $G$-martingale could not be form of ${\int }_0^t{\eta }_{s}ds$ or ${\int }_0^t{\gamma }_{s}d⟨B{⟩}_s$, $\eta ,\gamma \in M_G^1(0,T)$, which implies that the decomposition for generalized $G$-Itô processes is unique: For $\zeta \in H_G^1(0,T)$, $\eta \in M_G^1(0,T)$ 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 $\eta \equiv 0$, $\zeta \equiv 0$ and $K_t=L_t$. As an application, we give a characterization to the $G$-Sobolev spaces introduced in Peng and Song (2015).

References in corpus (2)

• Some properties on $G$-evaluation and its applications to $G$-martingale decomposition
• G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty