Existence and Stability of Solutions to Highly Nonlinear Stochastic Differential Delay Equations Driven by G-Brownian Motion
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摘要
Under linear expectation(or classical probability), the stability for stochastic differential delay equations(SDDEs), where their coeficients 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.
Under linear expectation(or classical probability), the stability for stochastic differential delay equations(SDDEs), where their coeficients 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.
Under linear expectation(or classical probability), the stability for stochastic differential delay equations(SDDEs), where their coeficients 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.
引文
[1] Z Chen. Strong laws of large numbers for sub-linear expectations, Sci China Math, 2016,59(5):945-954.
[2] Z Chen, P Wu, B Li. A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 2013, 54:365-377.
[3] X Bai, 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, 2014, 30:589-610.
[4] S Deng, C Fei, W Fei, X Mao. Stability equivalence between the stochastic differential delay equations driven by G-Brownian motion and the Euler-Maruyama method, Applied Mathematics Letters, 2019, 96:138-146.
[5] 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.
[6] 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).
[7] C Fei, W Fei, X Mao, M Shen, L Yan. Stability analysis of highly nonlinear hybrid multipledelay stochastic differential equations, Journal of Applied Analysis and Computations,2019, 9(3):1-18.
[8] C Fei, M Shen, W Fei, X Mao, L Yan. Stability of highly nonlinear hybrid stochastic integro-differential delay equations, Nonlinear Anal Hybrid Syst, 2019, 31:180–199.
[9] W Fei, C Fei. Optimal stochastic control and optimal consumption and portfolio with GBrownian motion, arXiv:1309.0209v1, 2013.
[10] W Fei, C Fei. On exponential stability for stochastic differential equations disturbed by G-Brownian motion, arXiv:1311.7311v1, 2013.
[11] W Fei, L Hu, X Mao, M Shen. Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 2017, 28:165-170.
[12] W Fei, L Hu, X Mao, M Shen. Structured robust stability and boundedness of nonlinear hybrid delay dystems, SIAM Contr Opt, 2018, 56:2662-2689.
[13] W Fei, L Hu, X Mao, M Shen. Generalised criteria on delay dependent stability of highly nonlinear hybrid stochastic systems, International Journal of Robust and Nonlinear Control, 2019, 25, 1201-1215.
[14] F Gao. Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stoch Proc Appl, 2009, 119(10):3356-3382.
[15] L Hu, X Mao, Y Shen. Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Systems&Control Letters, 2013, 62:178-187.
[16] X Li, X Lin, Y Lin. Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion, J Math Anal Appl, 2016, 439:235-255.
[17] X Li, S Peng. Stopping times and related It?os calculus with G-Brownian motion, Stoch Proc Appl, 2009, 121:1492-1508.
[18] Y Li, L Yan. Stability of delayed Hopfield neural networks under a sublinear expectation framework, Journal of the Franklin Institute, 2018, 355:4268-4281.
[19] Q Lin. Some properties of stochastic differential equations driven by G-Brownian motion,Acta Math Sin(Engl Ser), 2013, 29:923-942.
[20] Y Lin. Stochastic differential eqations driven by G-Brownian motion with reflecting boundary, Electron J Probab, 2013, 18(9):1-23.
[21] P Luo, F Wang. Stochastic differential equations driven by G-Brownian motion and ordinary differential equations, Stoch Proc Appl, 2014, 124:3869-3885.
[22] 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.
[23] X Mao. Stochastic Differential Equations and Their Applications, 2nd Edition, Chichester,Horwood Pub, 2007.
[24] X Mao, C Yuan. Stochastic Differential Equations with Markovian Swtching, Imperial College Press, London, 2006.
[25] S Peng. Nonlinear expectations and stochastic calculus under uncertainty, arXiv:1002.4546v1, 2010.
[26] Y Ren, X Jia, R Sakthivel. The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion, Applicable Analysis, 2017, 96(6):988-1003.
[27] Y Ren, W Yin, R Sakthivel. Stabilization of stochastic differential equations driven by G-Brownian motion with feedback control based on discrete-time state observation, Automatica, 2018, 95:146-151.
[28] B Shen, Z Wang, H Tan. Guaranteed cost control for uncertain nonlinear systems with mixed time-delays:The discrete-time case, European Journal of Control, 2018, 40:62-67.
[29] 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.
[30] 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.
[31] M Shen, W Fei, X Mao, Y Liang. Stability of highly nonlinear neutral stochastic differential delay equations, Systems&Control Letters, 2018, 115:1-8.
[32] L Zhang. Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Sci China Math, 2016, 59(4):751-768.
[33] D Zhang, Z Chen. Exponential stability for stochastic differential equation driven by GBrownian motion, Applied Mathematics Letters, 2012, 25:1906-1910.
[2] Z Chen, P Wu, B Li. A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 2013, 54:365-377.
[3] X Bai, 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, 2014, 30:589-610.
[4] S Deng, C Fei, W Fei, X Mao. Stability equivalence between the stochastic differential delay equations driven by G-Brownian motion and the Euler-Maruyama method, Applied Mathematics Letters, 2019, 96:138-146.
[5] 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.
[6] 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).
[7] C Fei, W Fei, X Mao, M Shen, L Yan. Stability analysis of highly nonlinear hybrid multipledelay stochastic differential equations, Journal of Applied Analysis and Computations,2019, 9(3):1-18.
[8] C Fei, M Shen, W Fei, X Mao, L Yan. Stability of highly nonlinear hybrid stochastic integro-differential delay equations, Nonlinear Anal Hybrid Syst, 2019, 31:180–199.
[9] W Fei, C Fei. Optimal stochastic control and optimal consumption and portfolio with GBrownian motion, arXiv:1309.0209v1, 2013.
[10] W Fei, C Fei. On exponential stability for stochastic differential equations disturbed by G-Brownian motion, arXiv:1311.7311v1, 2013.
[11] W Fei, L Hu, X Mao, M Shen. Delay dependent stability of highly nonlinear hybrid stochastic systems, Automatica, 2017, 28:165-170.
[12] W Fei, L Hu, X Mao, M Shen. Structured robust stability and boundedness of nonlinear hybrid delay dystems, SIAM Contr Opt, 2018, 56:2662-2689.
[13] W Fei, L Hu, X Mao, M Shen. Generalised criteria on delay dependent stability of highly nonlinear hybrid stochastic systems, International Journal of Robust and Nonlinear Control, 2019, 25, 1201-1215.
[14] F Gao. Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion, Stoch Proc Appl, 2009, 119(10):3356-3382.
[15] L Hu, X Mao, Y Shen. Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Systems&Control Letters, 2013, 62:178-187.
[16] X Li, X Lin, Y Lin. Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion, J Math Anal Appl, 2016, 439:235-255.
[17] X Li, S Peng. Stopping times and related It?os calculus with G-Brownian motion, Stoch Proc Appl, 2009, 121:1492-1508.
[18] Y Li, L Yan. Stability of delayed Hopfield neural networks under a sublinear expectation framework, Journal of the Franklin Institute, 2018, 355:4268-4281.
[19] Q Lin. Some properties of stochastic differential equations driven by G-Brownian motion,Acta Math Sin(Engl Ser), 2013, 29:923-942.
[20] Y Lin. Stochastic differential eqations driven by G-Brownian motion with reflecting boundary, Electron J Probab, 2013, 18(9):1-23.
[21] P Luo, F Wang. Stochastic differential equations driven by G-Brownian motion and ordinary differential equations, Stoch Proc Appl, 2014, 124:3869-3885.
[22] 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.
[23] X Mao. Stochastic Differential Equations and Their Applications, 2nd Edition, Chichester,Horwood Pub, 2007.
[24] X Mao, C Yuan. Stochastic Differential Equations with Markovian Swtching, Imperial College Press, London, 2006.
[25] S Peng. Nonlinear expectations and stochastic calculus under uncertainty, arXiv:1002.4546v1, 2010.
[26] Y Ren, X Jia, R Sakthivel. The p-th moment stability of solutions to impulsive stochastic differential equations driven by G-Brownian motion, Applicable Analysis, 2017, 96(6):988-1003.
[27] Y Ren, W Yin, R Sakthivel. Stabilization of stochastic differential equations driven by G-Brownian motion with feedback control based on discrete-time state observation, Automatica, 2018, 95:146-151.
[28] B Shen, Z Wang, H Tan. Guaranteed cost control for uncertain nonlinear systems with mixed time-delays:The discrete-time case, European Journal of Control, 2018, 40:62-67.
[29] 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.
[30] 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.
[31] M Shen, W Fei, X Mao, Y Liang. Stability of highly nonlinear neutral stochastic differential delay equations, Systems&Control Letters, 2018, 115:1-8.
[32] L Zhang. Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Sci China Math, 2016, 59(4):751-768.
[33] D Zhang, Z Chen. Exponential stability for stochastic differential equation driven by GBrownian motion, Applied Mathematics Letters, 2012, 25:1906-1910.