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

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

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.
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, (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.  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).   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.  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.  doi: 10.1016/j.spa.2013.09.010.  Google Scholar

[6]

K. Itô, On stochastic differential equations,, Mem. Amer. Math. Soc., 1951 (1951).   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.  doi: 10.1016/j.spa.2011.03.009.  Google Scholar

[8]

B. Øksendal, Stochastic Differential Equations. An Introduction with Applications,, Sixth edition, (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, (2007), 541.  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.  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.  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, (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.  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).   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.  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.  doi: 10.1016/j.spa.2013.09.010.  Google Scholar

[6]

K. Itô, On stochastic differential equations,, Mem. Amer. Math. Soc., 1951 (1951).   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.  doi: 10.1016/j.spa.2011.03.009.  Google Scholar

[8]

B. Øksendal, Stochastic Differential Equations. An Introduction with Applications,, Sixth edition, (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, (2007), 541.  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.  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.  doi: 10.1016/j.spa.2008.02.001.  Google Scholar

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