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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

H. Doss, Liens entre équations différentielles stochastiques et ordinaires, (French) C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A939-A942.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

K. Itô, On stochastic differential equations, Mem. Amer. Math. Soc., 1951 (1951), 51 pp.  Google Scholar

[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.  Google Scholar

[8]

B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Sixth edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

[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.  Google Scholar

[10]

S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and $G$-Brownian motion,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edition, Academic Press, Orlando, 1986.  Google Scholar

[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.  Google Scholar

[3]

H. Doss, Liens entre équations différentielles stochastiques et ordinaires, (French) C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A939-A942.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

K. Itô, On stochastic differential equations, Mem. Amer. Math. Soc., 1951 (1951), 51 pp.  Google Scholar

[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.  Google Scholar

[8]

B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Sixth edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

[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.  Google Scholar

[10]

S. Peng, Nonlinear expectations and stochastic calculus under uncertainty-with robust central limit theorem and $G$-Brownian motion,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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