Study of a Stochastic Differential System of Arbitrary Order under G-Brownian Motion.


Download PDF

Authors: E-H. CHALABI, A. OUAOUA AND S. MESBAHI

DOI: 10.46793/KgJMat2602.187C

Abstract:

In this paper, we study the existence and uniqueness of the solution for a class of stochastic differential systems of arbitrary order driven by G-Brownian motion. We prove under certain suitable conditions that our system has a unique solution. We also prove a stability theorem for our system.



Keywords:

G-expectation, G-Brownian motion, G-stochastic differential equations, G-Ito’s integral.



References:

[1]   X. Bai and Y. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with Integral-Lipschitz coefficients, Acta Math. Appl. Sin. Engl. Ser. 30(3) (2014), 589–610. https://doi.org/10.1007/s10255-014-0405-9

[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 (2011), 139–161. https://doi.org/10.1007/s11118-010-9185-x

[3]   F. Faizullah, A. Mukhtar and M. A. Rana, A note on stochastic functional differential equations driven by G-Brownian motion with discontinuous drift coefficients, Journal of Computational Analysis and Applications 21(5) (2016), 910–919.

[4]   F. Faizullah, W. A. Khan, M. Arif and R. A. Khan, On the Existence of Solutions for Stochastic Differential Equations under G-Brownian Motion, Life Science Journal 10(5) (2013), 255–260.

[5]   F. Gao, Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Process. Appl. 10(119) (2009), 3356–3382. https://doi.org/10.1016/j.spa.2009.05.010

[6]   M. Hu and X. Li, Independence under the G-expectation framework, J. Theoret. Probab. 27(3) (2014), 1011–1020. https://doi.org/10.1007/s10959-012-0471-y

[7]   Q. Lin, Some properties of stochastic differential equations driven by the G-Brownian motion, Acta Math. Sin. (Engl. Ser.) 29 (2013), 923–942. https://doi.org/10.1007/s11425-012-4534-4

[8]   Y. Lin, Stochastic differential equations driven by G-Brownian motion with reflecting boundary conditions, Electron. J. Probab. 18 (2013), 1–23. https://doi.org/10.1214/EJP.v18-2566

[9]   Y. Lin, Equations différentielles stochastiques sous les espérances mathématiques non-linéaires et applications, Yhèse Université de Rennes 1, 2013.

[10]   L. Peng and W. Falei, Stochastic differential equations driven by G-Brownian motion and ordinary differential equations, Stochastic Process. Appl. 124 (2014), 3869–3885. https://doi.org/10.1016/j.spa.2014.07.004

[11]   S. Peng, Nonlinear expectations and stochastic calculus under uncertainty with a new central limit theorem and G-Brownian motion, Institute of Mathematics, Shandong University, China, 2010. https://doi.org/10.48550/arXiv.1002.4546

[12]   S. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Science in China Series A Mathematics 52(7) (2009), 1391–1411. https://doi.org/10.1007/s11425-009-0121-8

[13]   S. Peng, G-Brownian motion and dynamic risk measure under volatility uncertainty, Institute of Mathematics, Shandong University, China, 2007. https://doi.org/10.48550/arXiv.0711.2834

[14]   S. Peng, G-expectation,G-Brownian motion and related stochastic calculus of Itô’s type, The Abel Symposium 2005, Abel Symposia 2, Springer-Verlag, 541–567, 2006. https://doi.org/10.1007/978-3-540-70847-6\_25

[15]   Y. Ren, W. Yin and D. Zhu, Stabilisation of SDEs and applications to synchronisation of stochastic neural network driven by G-Brownian motion with state-feedback control, Internat. J. Systems Sci. 50 (2019), 273–282. https://doi.org/10.1080/00207721.2018.1551973

[16]   Y. Ren, J. Wang and L. Hu, Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems, Internat. J. Control 90(5) (2017), 1132–1154. https://doi.org/10.1080/00207179.2016.1204560

[17]   A. Soumana-Hima, Stochastic differential equations under G-expectation and applications, Analysis of PDEs [math.AP]. Université Rennes 1, 2017. English. NNT : 2017REN1S007. tel-01527503. HAL ID:tel-01527503.

[18]   F. Yang, Harnack inequality and applications for SDEs driven by G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser. 36 (2020), 627–635.

[19]   D. Zhang and Z. Chen, Stability theorem for stochastic differential equations driven by G-Brownian motion, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat 19(3) (2011), 205–221.