Moderate deviations for neutral functional stochastic differential equations driven by Levy noises

Abstract

Using the weak convergence method introduced by A. Budhiraja, P. Dupuis, and A. Ganguly [Ann. Probab., 2016, 44: 1723{1775], we establish the moderate deviation principle for neutral functional stochastic differential equations driven by both Brownian motions and Poisson random measures.

Keywords

Moderate deviations / neutral functional stochastic dierential equations / Poisson random measure

References

[1] Aldous D. Stopping times and tightness. Ann Probab, 1978, 6: 335–340

[2] Bao J H, Yuan C G. Large deviations for neutral functional SDEs with jumps. Stochastics, 2015, 87: 48–70

[3] Budhiraja A, Chen J, Dupuis P. Large deviations for stochastic partial differential equations driven by a Poisson random measure. Stochastic Process Appl, 2013, 123: 523–560

[4] Budhiraja A, Dupuis P. A variational representation for positive functionals of infinite dimensional Brownian motion. Probab Math Statist, 2000, 20: 39–61

[5] Budhiraja A, Dupuis P, Ganguly A. Moderate deviation principle for stochastic differential equations with jumps. Ann Probab, 2016, 44: 1723–1775

[6] Budhiraja A, Dupuis P, Maroulas V. Variational representations for continuous time processes. Ann Inst Henri Poincaré Probab Stat, 2011, 47: 725–747

[7] Cai Y J, Huang J H, Maroulas V. Large deviations of mean-field stochastic differential equations with jumps. Statist Probab Lett, 2015, 96: 1–9

[8] Cerrai S, Röckner M. Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann Probab, 2004, 32: 1100–1139

[9] Dembo A, Zeitouni O, Large Deviations Techniques and Applications. San Diego: Academic Press, 1989

[10] Dong Z, Xiong J, Zhai J L, Zhang T. A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Levy noises. J Funct Anal, 2017, 272: 227–254

[11] Dunford N, Schwartz J. Linear Operators, Part I. New York: Wiley, 1988

[12] Dupuis P, Ellis R S. A Weak Convergence Approach to the Theory of Large Deviations. New York: Wiley, 1997

[13] Freidlin M I. Random perturbations of reaction-diffusion equations: the quasideterministic approach. Trans Amer Math Soc, 1988, 305: 665–697

[14] Freidlin M I, Wentzell A D. Random Perturbations of Dynamical Systems. New York: Springer, 1984

[15] Guillin A. Averaging principle of SDE with small diffusion: moderate deviations. Ann Probab, 2003, 31: 413–443

[16] Guillin A, Liptser R. Examples of moderate deviation principle for diffusion processes. Discrete Contin Dyn Syst Ser B, 2006, 6: 803–828

[17] He Q, Yin G. Large deviations for multi-scale Markovian switching systems with a small diffusion. Asymptot Anal, 2014, 87: 123–145

[18] He Q, Yin G. Moderate deviations for time-varying dynamic systems driven by nonhomogeneous Markov chains with two-time scales. Stochastics, 2014, 86: 527–550

[19] He Q, Yin G, Zhang Q. Large deviations for two-time-scale systems driven by nonhomogeneous Markov chains and associated optimal control problems. SIAM J Control Optim, 2011, 49: 1737–1765

[20] Kallianpur G, Xiong J. Large deviations for a class of stochastic partial differential equations. Ann Probab, 1996, 24: 320–345

[21] Ma X C, Xi F B. Moderate deviations for neutral stochastic differential delay equations with jumps. Statist Probab Lett, 2017, 126: 97–107

[22] Mao X. Stochastic Differential Equations and Applications. Amsterdam: Elsevier, 2007

[23] Maroulas V. Uniform large deviations for infinite dimensional stochastic systems with jumps. Mathematika, 2011, 57: 175–192

[24] Peszat S. Large derivation principle for stochastic evolution equations. Probab Theory Related Fields, 1994, 98: 113–136

[25] Röckner M, Zhang T, Zhang X. Large deviations for stochastic tamed 3D Navier-Stokes equations. Appl Math Optim, 2010, 61: 267–285

[26] Sowers R. Large deviations for a reaction-diffusion equation with non-Gaussian perturbations. Ann Probab, 1992, 20: 504–537

[27] Suo Y Q, Tao J, Zhang W. Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth. Front Math China, 2018, 13: 913–933

[28] Wang R, Zhai J L, Zhang T. A moderate deviation principle for 2-D stochastic Navier-Stokes equations. J Differential Equations, 2015, 258: 3363–3390

[29] Wang R, Zhang T. Moderate deviations for stochastic reaction-diffusion equations with multiplicative noise. Potential Anal, 2015, 42: 99–113

[30] Zhai J L, Zhang T. Large deviations for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises. Bernoulli, 2015, 21: 2351{2392

[31] Zhang X. Euler schemes and large deviations for stochastic Volterra equations with singular kernels. J Differential Equations, 2008, 244: 2226–2250

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