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

Yongsheng Song (Chinese Academy of Sciences, Beijing)


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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-15

Publication Date: March 19, 2012

DOI: 10.1214/EJP.v17-1890


  • Denis, Laurent; Hu, Mingshang; Peng, Shige. Function spaces and capacity related to a sublinear expectation: application to $G$-Brownian motion paths. Potential Anal. 34 (2011), no. 2, 139--161. MR2754968
  • Hu, Ming-shang; Peng, Shi-ge. On representation theorem of $G$-expectations and paths of $G$-Brownian motion. Acta Math. Appl. Sin. Engl. Ser. 25 (2009), no. 3, 539--546. MR2506990
  • Hu, Y. and Peng, S. Some Estimates for Martingale Representation under G-Expectation. arXiv:1004.1098v1.
  • Peng, Shige. $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type. Stochastic analysis and applications, 541--567, Abel Symp., 2, Springer, Berlin, 2007. MR2397805
  • Peng, S. G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty. arXiv:0711.2834v1.
  • Peng, Shige. Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation. Stochastic Process. Appl. 118 (2008), no. 12, 2223--2253. MR2474349
  • Peng, S. Nonlinear Expectations and Stochastic Calculus under Uncertainty, arXiv:1002.4546v1.
  • Peng, S, Song Y, Zhang J. A Complete Representation Theorem for G-martingales, arXiv:1201.2629v1.
  • Pham T, Zhang J. Some Norm Estimates for Semimartingales --Under Linear and Nonlinear Expectations, arXiv:1107.4020v1.
  • Soner, H. Mete; Touzi, Nizar; Zhang, Jianfeng. Martingale representation theorem for the $G$-expectation. Stochastic Process. Appl. 121 (2011), no. 2, 265--287. MR2746175
  • Song, YongSheng. Some properties on $G$-evaluation and its applications to $G$-martingale decomposition. Sci. China Math. 54 (2011), no. 2, 287--300. MR2771205
  • Song, Yongsheng. Properties of hitting times for $G$-martingales and their applications. Stochastic Process. Appl. 121 (2011), no. 8, 1770--1784. MR2811023
  • Song, Y. Characterizations of processes with stationary and independent increments under G-expectation, arXiv:1009.0109v1.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.