## Abstract and Applied Analysis

### Representation Theorem for Generators of BSDEs Driven by $G$-Brownian Motion and Its Applications

#### Abstract

We obtain a representation theorem for the generators of BSDEs driven by $G$-Brownian motions and then we use the representation theorem to get a converse comparison theorem for $G$-BSDEs and some equivalent results for nonlinear expectations generated by $G$-BSDEs.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 342038, 10 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393443619

Digital Object Identifier
doi:10.1155/2013/342038

Mathematical Reviews number (MathSciNet)
MR3147832

Zentralblatt MATH identifier
1310.60082

#### Citation

He, Kun; Hu, Mingshang. Representation Theorem for Generators of BSDEs Driven by $G$ -Brownian Motion and Its Applications. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 342038, 10 pages. doi:10.1155/2013/342038. https://projecteuclid.org/euclid.aaa/1393443619

#### References

• É. Pardoux and S. G. Peng, “Adapted solution of a backward stochastic differential equation,” Systems & Control Letters, vol. 14, no. 1, pp. 55–61, 1990.
• S. Peng, “Backward SDE and related $g$-expectation,” in Backward Stochastic Differential Equations, N. El Karoui and L. Mazliak, Eds., vol. 364 of Pitman Research Notes in Mathematics Series, pp. 141–159, 1997.
• Z. J. Chen, “A property of backward stochastic differential equations,” Comptes Rendus de l'Académie des Sciences. Série I, vol. 326, no. 4, pp. 483–488, 1998.
• P. Briand, F. Coquet, Y. Hu, J. Mémin, and S. G. Peng, “A converse comparison theorem for BSDEs and related properties of $g$-expectation,” Electronic Communications in Probability, vol. 5, pp. 101–117, 2000.
• L. Jiang, “Representation theorems for generators of backward stochastic differential equations,” Comptes Rendus Mathématique. Académie des Sciences. Paris I, vol. 340, no. 2, pp. 161–166, 2005.
• L. Jiang, “Converse comparison theorems for backward stochastic differential equations,” Statistics & Probability Letters, vol. 71, no. 2, pp. 173–183, 2005.
• L. Jiang, “Convexity, translation invariance and subadditivity for $g$-expectations and related risk measures,” The Annals of Applied Probability, vol. 18, no. 1, pp. 245–258, 2008.
• S. Peng, “Filtration consistent nonlinear expectations and evaluations of contingent claims,” Acta Mathematicae Applicatae Sinica, vol. 20, no. 2, pp. 191–214, 2004.
• S. Peng, “Nonlinear expectations and nonlinear Markov chains,” Chinese Annals of Mathematics B, vol. 26, no. 2, pp. 159–184, 2005.
• S. Peng, “$G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô type,” in Stochastic Analysis and Applications, vol. 2 of The Abel Symposium, pp. 541–567, Springer, Berlin, Germany, 2007.
• S. Peng, “G-Brownian motion and dynamic risk measure čommentComment on ref. [11?]: Please update the information of these references[11,13,15,16,18,20,21?], if possible.under volatility uncertainty,” http://arxiv.org/abs/0711.2834v1.
• S. Peng, “Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation,” Stochastic Processes and Their Applications, vol. 118, no. 12, pp. 2223–2253, 2008.
• S. Peng, “A new central limit theorem under sublinear expectations,” http://arxiv.org/abs/0803.2656v1.
• H. M. Soner, N. Touzi, and J. Zhang, “Wellposedness of second order backward SDEs,” Probability Theory and Related Fields, vol. 153, no. 1-2, pp. 149–190, 2012.
• M. S. Hu, S. L. Ji, S. G. Peng, and Y. S. Song, “Backward stochastic differential equations driven by G-brownian motion,” http://arxiv.org/abs/1206.5889v1.
• M. S. Hu, S. L. Ji, S. G. Peng, and Y. S. Song, “Comparison theorem, Feynman-Kac formula and Girsanov transformation for BSDEs driven by G-Brownian motion,” http://arxiv.org/abs/1212.5403v1.
• S. Peng, “G.\emphG-Brownian motion and dynamic risk measure under volatility uncertainty,” http://arxiv.org/abs/0711.2834v1.
• S. G. Peng, “Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation,” Stochastic Processes and their Applications, vol. 118, no. 12, pp. 2223–2253, 2008.
• S. G. Peng, “Nonlinear expectations and stochastic calculus under uncertainty,” http://arxiv.org/abs/1002.4546v1.
• Y. S. Song, “Some properties on $G$-evaluation and its applications to $G$-martingale decomposition,” Science China Mathematics, vol. 54, no. 2, pp. 287–300, 2011. \endinput