Functional solution about stochastic differential equation driven by G-Brownian motion

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January 2015, 20(1): 281-293. doi: 10.3934/dcdsb.2015.20.281

Functional solution about stochastic differential equation driven by $G$-Brownian motion

Defei Zhang ^1, and Ping He ^1,

1.	Department of Mathematics, Honghe University, Mengzi, 661199, China, China

Received February 2013 Revised June 2013 Published November 2014

Abstract
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Peng introduced the notions of $G$-expectation and $G$-Brownian motion as well as $G$-Itô formula in 2006. The $G$-Brownian motion has many rich and new properties comparing to classical Brownian motion. In this paper, we present a method to solve stochastic differential equation driven by $G$-Brownian motion without using $G$-Itô formula. Our method is mainly depending on Frobenius's Theorem. Many classical models in mathematical finance are investigated to illustrate the method. As a by-product, this financial models are extended to the case of $G$-Brownian motion.

Keywords: $G$-stochastic differential equations, $G$-Itô formula., $G$-Brownian motion, Partial differential equation.

Mathematics Subject Classification: Primary: 60H10; Secondary: 60H3.

Citation: Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 281-293. doi: 10.3934/dcdsb.2015.20.281

References:

[1]	W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edition, Academic Press, Orlando, 1986.
[2]	L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths, Potential Anal., 34 (2010), 139-161. doi: 10.1007/s11118-010-9185-x.
[3]	H. Doss, Liens entre équations différentielles stochastiques et ordinaires, (French) C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A939-A942.
[4]	F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion, Stoch. Proc. Appl., 119 (2009), 3356-3382. doi: 10.1016/j.spa.2009.05.010.
[5]	M. Hu, S. Ji, S. Peng and Y. Song, Backward stochastic differential equations driven by $G$-Brownian motion, Stochastic Process. Appl., 124 (2014), 759-784. doi: 10.1016/j.spa.2013.09.010.
[6]	K. Itô, On stochastic differential equations, Mem. Amer. Math. Soc., 1951 (1951), 51 pp.
[7]	X. Li and S. Peng, Stopping times and related Itô's calculus with $G$-Brownian motion, Stoch. Proc. Appl., 121 (2011), 1492-1508. doi: 10.1016/j.spa.2011.03.009.
[8]	B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Sixth edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.
[9]	S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô's type, in Stochastic Analysis and Applications (eds. Benth, et al.), Springer-Verlag, 2007, 541-567. doi: 10.1007/978-3-540-70847-6_25.
[10]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and $G$-Brownian motion, preprint, arXiv:1002.4546v1.
[11]	H. M. Soner, N. Touzi and J. Zhang, Martingale representation theorem for the $G$-expectation, Stoch. Proc. Appl., 121 (2011), 265-287. doi: 10.1016/j.spa.2010.10.006.
[12]	J. Xu and B. Zhang, Martingale characterization of $G$-Brownian motion, Stoch. Proc. Appl., 119 (2009), 232-248. doi: 10.1016/j.spa.2008.02.001.

show all references

References:

[1]	W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edition, Academic Press, Orlando, 1986.
[2]	L. Denis, M. Hu and S. Peng, Function spaces and capacity related to a sublinear expectation: Application to $G$-Brownian motion paths, Potential Anal., 34 (2010), 139-161. doi: 10.1007/s11118-010-9185-x.
[3]	H. Doss, Liens entre équations différentielles stochastiques et ordinaires, (French) C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A939-A942.
[4]	F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by $G$-Brownian motion, Stoch. Proc. Appl., 119 (2009), 3356-3382. doi: 10.1016/j.spa.2009.05.010.
[5]	M. Hu, S. Ji, S. Peng and Y. Song, Backward stochastic differential equations driven by $G$-Brownian motion, Stochastic Process. Appl., 124 (2014), 759-784. doi: 10.1016/j.spa.2013.09.010.
[6]	K. Itô, On stochastic differential equations, Mem. Amer. Math. Soc., 1951 (1951), 51 pp.
[7]	X. Li and S. Peng, Stopping times and related Itô's calculus with $G$-Brownian motion, Stoch. Proc. Appl., 121 (2011), 1492-1508. doi: 10.1016/j.spa.2011.03.009.
[8]	B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Sixth edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.
[9]	S. Peng, $G$-expectation, $G$-Brownian motion and related stochastic calculus of Itô's type, in Stochastic Analysis and Applications (eds. Benth, et al.), Springer-Verlag, 2007, 541-567. doi: 10.1007/978-3-540-70847-6_25.
[10]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and $G$-Brownian motion, preprint, arXiv:1002.4546v1.
[11]	H. M. Soner, N. Touzi and J. Zhang, Martingale representation theorem for the $G$-expectation, Stoch. Proc. Appl., 121 (2011), 265-287. doi: 10.1016/j.spa.2010.10.006.
[12]	J. Xu and B. Zhang, Martingale characterization of $G$-Brownian motion, Stoch. Proc. Appl., 119 (2009), 232-248. doi: 10.1016/j.spa.2008.02.001.

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