Electronic Journal of Probability

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

Yongsheng Song

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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.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 24, 15 pp.

Accepted: 19 March 2012
First available in Project Euclid: 4 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G48: Generalizations of martingales
Secondary: 60G44: Martingales with continuous parameter

uniqueness representation theorem $G$-martingale finite variation $G$-expectation

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Song, Yongsheng. Uniqueness of the representation for $G$-martingales with finite variation. Electron. J. Probab. 17 (2012), paper no. 24, 15 pp. doi:10.1214/EJP.v17-1890. https://projecteuclid.org/euclid.ejp/1465062346

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