## Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

### Characterizations of processes with stationary and independent increments under $G$-expectation

Yongsheng Song

#### Abstract

Our purpose is to investigate properties for processes with stationary and independent increments under $G$-expectation. As applications, we prove the martingale characterization of $G$-Brownian motion and present a pathwise decomposition theorem for generalized $G$-Brownian motion.

#### Résumé

Notre but est d’étudier des propriétés de processus à accroissements stationnaires et indépendants sous une $G$-espérance. Comme application, nous démontrons la caractérisation de la martingale de $G$-mouvement Brownien et fournissons un théorème de décomposition trajectorielle pour le $G$-mouvement Brownien généralisé.

#### Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 252-269.

Dates
First available in Project Euclid: 29 January 2013

https://projecteuclid.org/euclid.aihp/1359470134

Digital Object Identifier
doi:10.1214/12-AIHP492

Mathematical Reviews number (MathSciNet)
MR3060156

Zentralblatt MATH identifier
1282.60050

#### Citation

Song, Yongsheng. Characterizations of processes with stationary and independent increments under $G$-expectation. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 252--269. doi:10.1214/12-AIHP492. https://projecteuclid.org/euclid.aihp/1359470134

#### References

• [1] L. Denis, M. Hu and S. Peng. Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion pathes. Potential Anal. 34 (2011) 139–161.
• [2] M. Hu and S. Peng. On representation theorem of $G$-expectations and paths of $G$-Brownian motion. Acta Math. Appl. Sin. Engl. Ser. 25 (2009) 539–546.
• [3] S. Peng. $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type. In Stochastic Analysis and Applications 541–567. Abel Symp. 2. Springer, Berlin, 2007.
• [4] S. Peng. $G$-Brownian motion and dynamic risk measure under volatility uncertainty. Available at arXiv:0711.2834v1 [math.PR], 2007.
• [5] S. Peng. Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation. Stochastic Process. Appl. 118 (2008) 2223–2253.
• [6] S. Peng. A new central limit theorem under sublinear expectations. Available at arXiv:0803.2656v1 [math.PR], 2008.
• [7] S. Peng. Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China Ser. A 52 (2009) 1391–1411.
• [8] S. Peng. Nonlinear expectations and stochastic calculus under uncertainty. Available at arXiv:1002.4546v1 [math.PR], 2010.
• [9] M. Soner, N. Touzi and J. Zhang. Martingale representation theorem under $G$-expectation. Stochastic Process. Appl. 121 (2011) 265–287.
• [10] Y. Song. Some properties on $G$-evaluation and its applications to $G$-martingale decomposition. Sci. China Math. 54 (2011) 287–300.
• [11] Y. Song. Properties of hitting times for $G$-martingales and their applications. Stochastic Process. Appl. 121 (2011) 1770–1784.
• [12] Y. Song. Uniqueness of the representation for $G$-martingales with finite variation. Electron. J. Probab. 17 (2012) 1–15.
• [13] J. Xu and B. Zhang. Martingale characterization of $G$-Brownian motion. Stochastic Process. Appl. 119 (2009) 232–248.