Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion
- 20 Downloads
Under linear expectation (or classical probability), the stability for stochastic differential delay equations (SDDEs), where their coefficients are either linear or nonlinear but bounded by linear functions, has been investigated intensively. Recently, the stability of highly nonlinear hybrid stochastic differential equations is studied by some researchers. In this paper, by using Peng’s G-expectation theory, we first prove the existence and uniqueness of solutions to SDDEs driven by G-Brownian motion (G-SDDEs) under local Lipschitz and linear growth conditions. Then the second kind of stability and the dependence of the solutions to G-SDDEs are studied. Finally, we explore the stability and boundedness of highly nonlinear G-SDDEs.
Keywordsstochastic differential delay equation (SDDE) sublinear expectation existence and uniqueness G-Brownian motion stability and boundedness
MR Subject Classification60H10 93E15
Unable to display preview. Download preview PDF.
- L Denis, M Hu, S Peng. Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths, arXiv:0802.1240, 2008.Google Scholar
- C Fei, W Fei. Consistency of least squares estimation to the parameter for stochastic differential equations under distribution uncertainty, Acta Math. Sci., Chinese Ser. (2019) accepted (see arXiv:1904.12701v1).Google Scholar
- W Fei, C Fei. Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion, arXiv:1309.0209v1, 2013.Google Scholar
- W Fei, C Fei. On exponential stability for stochastic differential equations disturbed by G-Brownian motion, arXiv:1311.7311v1, 2013.Google Scholar
- L Ma, Z Wang, R Liu, F Alsaadi. A note on guaranteed cost control for nonlinear stochastic systems with input saturation and mixed time-delays, International Journal of Robust and Nonlinear Control, 2017, 27 (18): 4443–4456.Google Scholar
- X Mao. Stochastic Differential Equations and Their Applications, 2nd Edition, Chichester, Horwood Pub, 2007.Google Scholar
- S Peng. Nonlinear expectations and stochastic calculus under uncertainty, arX-iv:1002.4546v1, 2010.Google Scholar
- M Shen, C Fei, W Fei, X Mao. The boundedness and stability of highly nonlinear hybrid neutral stochastic systems with multiple delays, Sci China Inf Sci, DOI: 10.1007/s11432-018-9755-7.Google Scholar
- M Shen, W Fei, X Mao, S Deng. Exponential stability of highly nonlinear neutral pantograph stochastic differential equations, Asian J Control, DOI: 10.1002/asjc.1903.Google Scholar