American Institute of Mathematical Sciences

Advanced
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

[1]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157
[2]

Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325
[3]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[4]

Mahmoud Abouagwa, Ji Li. G-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1583-1606. doi: 10.3934/dcdsb.2019241
[5]

Zhengyan Lin, Li-Xin Zhang. Convergence to a self-normalized G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 4-. doi: 10.1186/s41546-017-0013-8
[6]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[7]

Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete & Continuous Dynamical Systems - B, 2022, 27 (2) : 883-901. doi: 10.3934/dcdsb.2021072
[8]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[9]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[10]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[11]

Yong Ren, Wensheng Yin. Quasi sure exponential stabilization of nonlinear systems via intermittent $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5871-5883. doi: 10.3934/dcdsb.2019110
[12]

Yong Ren, Wensheng Yin, Dongjin Zhu. Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3347-3360. doi: 10.3934/dcdsb.2018248
[13]

Francesca Biagini, Thilo Meyer-Brandis, Bernt Øksendal, Krzysztof Paczka. Optimal control with delayed information flow of systems driven by G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 8-. doi: 10.1186/s41546-018-0033-z
[14]

Sel Ly, Nicolas Privault. Stochastic ordering by g-expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 61-98. doi: 10.3934/puqr.2021004
[15]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084
[16]

Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565
[17]

François Lalonde, Egor Shelukhin. Proof of the main conjecture on $g$-areas. Electronic Research Announcements, 2015, 22: 92-102. doi: 10.3934/era.2015.22.92
[18]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[19]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[20]

Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $. Evolution Equations & Control Theory, 2021, 10 (4) : 921-935. doi: 10.3934/eect.2020096

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (189)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy