## Electronic Journal of Probability

### Supermartingale decomposition theorem under $G$-expectation

#### Abstract

The objective of this paper is to establish the decomposition theorem for supermartingales under the $G$-framework. We first introduce a $g$-nonlinear expectation via a kind of $G$-BSDE and the associated supermartingales. We have shown that this kind of supermartingales has the decomposition similar to the classical case. The main ideas are to apply the property on uniform continuity of $S_G^\beta (0,T)$, the representation of the solution to $G$-BSDE and the approximation method via penalization.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 50, 20 pp.

Dates
Accepted: 28 April 2018
First available in Project Euclid: 1 June 2018

https://projecteuclid.org/euclid.ejp/1527818428

Digital Object Identifier
doi:10.1214/18-EJP173

Mathematical Reviews number (MathSciNet)
MR3814244

Zentralblatt MATH identifier
06924662

#### Citation

Li, Hanwu; Peng, Shige; Song, Yongsheng. Supermartingale decomposition theorem under $G$-expectation. Electron. J. Probab. 23 (2018), paper no. 50, 20 pp. doi:10.1214/18-EJP173. https://projecteuclid.org/euclid.ejp/1527818428

#### References

• [1] Bensoussan, A.: On the theory of option pricing. Acta. Appl. Math. 2, (1984), 139–158.
• [2] Briand, P., Delyon, B., Hu, Y., Pardoux, E. and Stoica, L.: $L^p$ solutions of backward stochastic differential equations. Stochastic Processes and their Applications 108, (2003), 109–129.
• [3] Chen, Z. and Peng, S.: Continuous properties of $g$-martingales. Chin. Ann. of Math. 22B, (2001), 115–128.
• [4] Coquet, F., Hu, Y., Mémin, J. and Peng, S.: Filtration-consistent nonlinear expectations and related g-expectations. Probab. Theory Relat. Fields 123, (2002), 1–27.
• [5] Crandall, M. G., Ishii, H. and Lions, P. L.: User’s guide to viscosity solutions of second order partial differential equations. Bulletin of The American Mathematical Society 27(1), (1992), 1–67.
• [6] Cvitanic, J. and Karatzas, I.: Hedging contingent claims with constrained portfolios. Ann. of Appl. Proba. 3(3), (1993), 652–681.
• [7] Delbaen, F., Peng, S. and Rosazza, G. E.: Representation of the penalty term of dynamic concave utilities. Finance and Stochastics 14, (2010), 449–472.
• [8] Denis, L., Hu, M. and Peng S.: Function spaces and capacity related to a sublinear expectation: application to $G$-Brownian motion pathes. Potential Anal. 34, (2011), 139–161.
• [9] Doob, L.: Stochastic Process. John Wiley & Sons, New York, NY, USA, (1953).
• [10] Hu, M., Ji, S., Peng, S. and Song, Y.: Backward stochastic differential equations driven by $G$-Brownian motion. Stochastic Processes and their Applications 124, (2014), 759–784.
• [11] Hu, M., Ji, S., Peng, S. and Song, Y.: Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by $G$-Brownian motion. Stochastic Processes and their Applications 124, (2014), 1170–1195.
• [12] Hu, M. and Peng, S.: On representation theorem of $G$-expectations and paths of $G$-Brownian motion. Acta Math. Appl. Sin. Engl. Ser. 25(3), (2009), 539–546.
• [13] Hu, M. and Peng, S.: Extended conditional $G$-expectations and related stopping times, arXiv:1309.3829v1
• [14] Karatzas, I.: On the pricing of American options. Applied Mathematics and Optimization 17, (1988), 37–60.
• [15] Li, X. and Peng, S.: Stopping times and related Itô’s calculus with $G$-Brownian motion. Stochastic Processes and their Applications 121, (2011), 1492–1508.
• [16] Meyer, P. A.: A decomposition theorem for supermartingales. Illinois J. Math. 6, (1962), 193–205.
• [17] Meyer, P. A.: Decomposition for supermartingales: the uniqueness theorem. Illinois J. Math. 7, (1963), 1–17.
• [18] Pardoux, E. and Peng, S.: Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14, (1990), 55–61.
• [19] Peng, S.: BSDE and related g-expectations. In El Karoui, N., Mazliak, L. (eds.) Backward Stochastic Differential Equations, No. 364 in Pitman Research Notes in Mathematics Series, Addison Wesley, Longman, London, (1997), 141–159.
• [20] Peng, S.: Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. Probab. Theory Relat. Fields. 113, (1999), 473–499.
• [21] Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures, in: Lectures Notes in CIME-EMS Summer School, 2003, Bressanone, in: Springer’s Lecture Notes in Mathematics, vol. 1856.
• [22] Peng, S.: Dynamically consistent nonlinear evaluations and expectations, arXiv:math/0501415v1
• [23] Peng, S.: $G$-expectation, $G$-Brownian Motion and Related Stochastic Calculus of Itô type. Stochastic analysis and applications (2007) 541–567, Abel Symp., 2, Springer, Berlin.
• [24] Peng, S.: Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation. Stochastic Processes and their Applications 118(12), (2008), 2223–2253.
• [25] Peng, S.: Nonlinear expectations and stochastic calculus under uncertainty, arXiv:1002.4546v1
• [26] Peng, S., Song, Y. and Zhang, J. A complete representation theorem for G-martingales. Stochastics 86(4), (2014), 609–631.
• [27] Pham, T. and Zhang, J.: Some norm estimates for semimaringales. Electron. J. Probab. 18, (2013), 1–25.
• [28] Soner, M., Touzi, N. and Zhang, J.: Martingale representation theorem under G-expectation. Stochastic Processes and Their Applications. 121, (2011), 265–287.
• [29] Soner, M., Touzi, N. and Zhang, J.: Wellposedness of second order backward SDEs. Probab. Theory Related Fields 153, (2012), 149–190.
• [30] Song, Y.: Some properties on G-evaluation and its applications to G-martingale decomposition. Science China Mathematics 54, (2011), 287–300.
• [31] Song, Y.: Uniqueness of the representation for G-martingales with finite variation. Electron. J. Probab 17, (2012), 1-15.