Uniqueness of the representation for $G$-martingales with finite variation
Uniqueness of the representation for $G$-martingales with finite variation
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DOI：
10.1214/ejp.v17-1890
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发表时间：
2010-12
影响因子：
1.4
通讯作者：
Yongsheng Song
中科院分区：
文献类型：
--
作者：
Yongsheng Song
设$\{\delta_n\}$是区间$[0，T]$上的Rademacher函数的精化序列，通过[d（K）=\limsup_n\hat{E}[\int_0^T\delta_n（s）dK_s]在$G$-期望空间中引入一个过程泛函.本文证明了如果$K_t=\int_0^t\eta_sd\langle B\rangle_s$且$\eta\在M^1_G（0，T）$中是非平凡的，则$d（K）>0$;如果$K_t=\int_0^t\eta_sds$且$\eta\在M^1_G（0，T）$中是非平凡的，则$d（K）=0$.这意味着有限变差的$G$-鞅的表示的唯一性，这是本文的主要目的。
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.