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Applied Mathematics-A Journal of Chinese Universities  2019, Vol. 34 Issue (2): 184-    DOI: 10.1007/s11766-019-3619-x
    
Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion
FEI Chen, FEI Wei-yin, YAN Li-tan
1 Glorious Sun School of Business and Management,Donghua University, Shanghai, 200051,China.
2 School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, Anhui 241000,China.
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Abstract  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.

Key words： stochastic differential delay equation (SDDE)      sublinear expectation      existence and uniqueness      G-Brownian motion, stability and boundedness     
Published: 03 July 2019
CLC:  60H10  
  93E15  
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FEI Chen
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FEI Chen, FEI Wei-yin, YAN Li-tan. Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion. Applied Mathematics-A Journal of Chinese Universities, 2019, 34(2): 184-.

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http://www.zjujournals.com/amjcub/10.1007/s11766-019-3619-x     OR     http://www.zjujournals.com/amjcub/Y2019/V34/I2/184


Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion

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.

关键词： stochastic differential delay equation (SDDE),  sublinear expectation,  existence and uniqueness,  G-Brownian motion, stability and boundedness 
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