Electronic Journal of Probability

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

Yongsheng Song

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062346

Digital Object Identifier
doi:10.1214/EJP.v17-1890

Mathematical Reviews number (MathSciNet)
MR2900465

Zentralblatt MATH identifier
1244.60046

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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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


Export citation

References

  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Song, YongSheng. Some properties on $G$-evaluation and its applications to $G$-martingale decomposition. Sci. China Math. 54 (2011), no. 2, 287–300.
  • Song, Yongsheng. Properties of hitting times for $G$-martingales and their applications. Stochastic Process. Appl. 121 (2011), no. 8, 1770–1784.
  • Song, Y. Characterizations of processes with stationary and independent increments under G-expectation, arXiv:1009.0109v1.