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April  2020, 19(4): 2369-2384. doi: 10.3934/cpaa.2020103

Large deviations for neutral stochastic functional differential equations

Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, UK

*Corresponding author

Dedicated to Professor Tomás Caraballo on occasion of his 60th Birthday

Received  March 2019 Revised  June 2019 Published  January 2020

In this paper, under a one-sided Lipschitz condition on the drift coefficient we adopt (via contraction principle) an exponential approximation argument to investigate large deviations for neutral stochastic functional differential equations.

Citation: Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103
References:
[1]

J. Bao, G. Yin and C. Yuan, Asymptotic Analysis for Functional Stochastic Differential Equations, Springer, Cham, 2016. doi: 10.1007/978-3-319-46979-9.  Google Scholar

[2]

J. Bao and C. Yuan, Large deviations for neutral functional SDEs with jumps, Stochastic, 87 (2015), 48-70.  doi: 10.1080/17442508.2014.914516.  Google Scholar

[3]

Li. Bo and T. Zhang, Large deviations for perturbed reflected diffusion processes, Stochastics, 81 (2009), 531-543.  doi: 10.1080/17442500801981084.  Google Scholar

[4]

A. BudhirajaJ. Chen and P. Dupuis, Large deviations for stochastic partial differential equations driven by Poisson random measure, Stochastic Process. Appl., 123 (2013), 523-560.  doi: 10.1016/j.spa.2012.09.010.  Google Scholar

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A. BudhirajaP. Dupuis and M. Fischer, Large deviation properties of weakly interacting processes via weak convergence methods, Ann. Probab., 40 (2012), 74-102.  doi: 10.1214/10-AOP616.  Google Scholar

[6]

A. BudhirajaP. Dupuis and A. Ganguly, Large deviations for small noise diffusions in a fast Markovian environment, Election. J. Probab., 23 (2018), 1-33.  doi: 10.1214/18-EJP228.  Google Scholar

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A. Budhiraja and P. Nyquist, Large deviations for multidimensional state-dependent shot-noise processes, J. Appl. Probab., 52 (2015), 1097-1114.  doi: 10.1239/jap/1450802755.  Google Scholar

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A. Dembo and A. Zeitouni, Large Deviations Techniques and Applications, Springer-Verlag, Berlin Heidelberg, 1998. doi: 10.1007/978-1-4612-5320-4.  Google Scholar

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M. Freidlin, Random perturbations of reaction-diffusion equations: the quasi-deterministic approximation, Trans. Amer. Math. Soc., 305 (1988), 665-697.  doi: 10.2307/2000884.  Google Scholar

[10]

S. Gadat, F. Panloup and C. Pellegrini, Large deviation principle for invariant distributions of memory gradient diffusions, Electron. J. Probab., 18 (2013), 34pp. doi: 10.1214/EJP.v18-2031.  Google Scholar

[11]

G. HuangM. Mandjes and P. Spreij, Large deviations for Markov-modulated diffusion processes with rapid switching, Stochastic Process. Appl., 126 (2016), 1785-1818.  doi: 10.1016/j.spa.2015.12.005.  Google Scholar

[12]

R. S. Liptser and A. A. Pukhalskii, Limit theorems on large deviations for semimartingales, Stochastics Stochastics Rep., 38 (1992), 201-249.  doi: 10.1080/17442509208833757.  Google Scholar

[13]

K. Liu and T. Zhang, A large deviation principle of retarded Ornstein-Uhlenbeck processes driven by Levy noise, Stoch. Anal. Appl., 32 (2014), 889-910.  doi: 10.1080/07362994.2014.939544.  Google Scholar

[14]

X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[15]

C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal., 80 (2013), 202-210.  doi: 10.1016/j.na.2012.10.004.  Google Scholar

[16]

S. A. Mohammed and T. Zhang, Large deviations for stochastic systems with memory, Discrete Contin. Dyn. Syst. Ser. B, 66 (2006), 881–893. doi: 10.3934/dcdsb.2006.6.881.  Google Scholar

[17]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles, Potential Anal., 26 (2007), 255-279.  doi: 10.1007/s11118-006-9035-z.  Google Scholar

[18]

D.W. Stroock, An Introduction to the Theory of Large Deviations, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-1-4613-8514-1.  Google Scholar

[19]

Y. SuoJ. Tao and W. Zhang, Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth, Front. Math. China, 13 (2018), 913-933.  doi: 10.1007/s11464-018-0710-3.  Google Scholar

show all references

References:
[1]

J. Bao, G. Yin and C. Yuan, Asymptotic Analysis for Functional Stochastic Differential Equations, Springer, Cham, 2016. doi: 10.1007/978-3-319-46979-9.  Google Scholar

[2]

J. Bao and C. Yuan, Large deviations for neutral functional SDEs with jumps, Stochastic, 87 (2015), 48-70.  doi: 10.1080/17442508.2014.914516.  Google Scholar

[3]

Li. Bo and T. Zhang, Large deviations for perturbed reflected diffusion processes, Stochastics, 81 (2009), 531-543.  doi: 10.1080/17442500801981084.  Google Scholar

[4]

A. BudhirajaJ. Chen and P. Dupuis, Large deviations for stochastic partial differential equations driven by Poisson random measure, Stochastic Process. Appl., 123 (2013), 523-560.  doi: 10.1016/j.spa.2012.09.010.  Google Scholar

[5]

A. BudhirajaP. Dupuis and M. Fischer, Large deviation properties of weakly interacting processes via weak convergence methods, Ann. Probab., 40 (2012), 74-102.  doi: 10.1214/10-AOP616.  Google Scholar

[6]

A. BudhirajaP. Dupuis and A. Ganguly, Large deviations for small noise diffusions in a fast Markovian environment, Election. J. Probab., 23 (2018), 1-33.  doi: 10.1214/18-EJP228.  Google Scholar

[7]

A. Budhiraja and P. Nyquist, Large deviations for multidimensional state-dependent shot-noise processes, J. Appl. Probab., 52 (2015), 1097-1114.  doi: 10.1239/jap/1450802755.  Google Scholar

[8]

A. Dembo and A. Zeitouni, Large Deviations Techniques and Applications, Springer-Verlag, Berlin Heidelberg, 1998. doi: 10.1007/978-1-4612-5320-4.  Google Scholar

[9]

M. Freidlin, Random perturbations of reaction-diffusion equations: the quasi-deterministic approximation, Trans. Amer. Math. Soc., 305 (1988), 665-697.  doi: 10.2307/2000884.  Google Scholar

[10]

S. Gadat, F. Panloup and C. Pellegrini, Large deviation principle for invariant distributions of memory gradient diffusions, Electron. J. Probab., 18 (2013), 34pp. doi: 10.1214/EJP.v18-2031.  Google Scholar

[11]

G. HuangM. Mandjes and P. Spreij, Large deviations for Markov-modulated diffusion processes with rapid switching, Stochastic Process. Appl., 126 (2016), 1785-1818.  doi: 10.1016/j.spa.2015.12.005.  Google Scholar

[12]

R. S. Liptser and A. A. Pukhalskii, Limit theorems on large deviations for semimartingales, Stochastics Stochastics Rep., 38 (1992), 201-249.  doi: 10.1080/17442509208833757.  Google Scholar

[13]

K. Liu and T. Zhang, A large deviation principle of retarded Ornstein-Uhlenbeck processes driven by Levy noise, Stoch. Anal. Appl., 32 (2014), 889-910.  doi: 10.1080/07362994.2014.939544.  Google Scholar

[14]

X. Mao, Stochastic Differential Equations and Applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.  Google Scholar

[15]

C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal., 80 (2013), 202-210.  doi: 10.1016/j.na.2012.10.004.  Google Scholar

[16]

S. A. Mohammed and T. Zhang, Large deviations for stochastic systems with memory, Discrete Contin. Dyn. Syst. Ser. B, 66 (2006), 881–893. doi: 10.3934/dcdsb.2006.6.881.  Google Scholar

[17]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles, Potential Anal., 26 (2007), 255-279.  doi: 10.1007/s11118-006-9035-z.  Google Scholar

[18]

D.W. Stroock, An Introduction to the Theory of Large Deviations, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-1-4613-8514-1.  Google Scholar

[19]

Y. SuoJ. Tao and W. Zhang, Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth, Front. Math. China, 13 (2018), 913-933.  doi: 10.1007/s11464-018-0710-3.  Google Scholar

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