Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

Abstract

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coeffcients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coeffcients are polynomial growth with respect to the delay variables.

Keywords

Stochastic differential delay equation (SDDE) / polynomial growth / central limit theorem / moderate deviation principle / weak convergence

References

[1] Bao J, Wang F, Yuan C. Ergodicity for neutral type SDEs with infinite memory. arXiv: 1805.03431

[2] Bao J, Yuan C. Convergence rate of EM scheme for SDDEs. Proc Amer Math Soc, 2013, 141(9): 3231–3243

[3] Bao J, Yuan C. Large deviations for neutral functional SDEs with jumps. Stochastics, 2015, 87(1): 48–70

[4] Budhiraja A, Chen, J, Dupuis P. Large deviations for stochastic partial differential equations driven by Poisson random measure. Stochastic Process Appl, 2013, 123(2): 523–560

[5] Budhiraja A, Dupuis P. A variational representation for positive functionals of infinite dimensional Brownian motion. Probab Math Statist, 2000, 20(1): 39–61

[6] Budhiraja A, Dupuis P, Fischer M. Large deviation properties of weakly interacting processes via weak convergence methods. Ann Probab, 2012, 40(1): 74–102

[7] Budhiraja A, Dupuis P, Ganguly A. Moderate deviation principles for stochastic differential equations with jumps. Ann Probab, 2016, 44(3): 1723–1775

[8] Budhiraja A, Dupuis P, Maroulas V. Large deviations for infinite dimensional stochastic dynamical systems. Ann Probab, 2008, 36(4): 1390–1420

[9] Budhiraja A, Dupuis P, Maroulas V. Variational representations for continuous time processes. Ann Inst Henri Poincaré Probab Stat, 2011, 47(3): 725–747

[10] Budhiraja A, Nyquist P. Large deviations for multidimensional state-dependent shotnoise processes. J Appl Probab, 2015, 52(4): 1097–1114

[11] Cardon-Webber C. Large deviations for Burger's-type SPDE. Stochastic Process Appl, 1999, 84(1): 53–70

[12] Cerrai S, Röckner M. Large deviations for stochastic reaction diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann Probab, 2004, 32(18): 1100–1139

[13] Chow P. Large deviation problem for some parabolic Itô equations. Comm Pure Appl Math, 1992, 45(1): 97–120

[14] Cortissoz J. On the Skorokhod representation theorem. Proc Amer Math Soc, 2007, 135(12): 3995–4007

[15] Dembo A, Zeitouni O. Large Deviations Techniques and Applications.2nd ed. Applications of Mathematics, Vol 38. Berlin: Springer-Verlag, 1998

[16] Duan J, Millet A. Large deviations for the Boussinesq equations under random inuences. Stochastic Process Appl, 2009, 119(6): 2052–2081

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

[18] Fatheddin P. Central limit theorem for a class of SPDEs. J Appl Probab, 2015, 52(3): 786–796

[19] Freidlin M. Random perturbations of reaction-diffusion equations: the quasideterministic approximation. Trans Amer Math Soc, 1988, 305(2): 665–697

[20] Freidlin M, Wentzell A. Random Perturbation of Dynamical Systems. Grundlehren Math Wiss, Vol 260. New York: Springer, 1984

[21] Horbacz K. The central limit theorem for random dynamical systems. J Stat Phys, 2016, 164(6): 1261–1291

[22] Kallianpur G, Xiong J. Large deviations for a class of stochastic partial differential equations. Ann Probab, 1996, 24(1): 320–345

[23] Karatzas I, Shreve S. Brownian Motion and Stochastic Calculus.2nd ed. Grad Texts in Math, Vol 113. New York: Springer-Verlag, 1991

[24] Klebaner F C. Introduction to Stochastic Calculus with Applications.2nd ed. London: Imperial College Press, 2005

[25] Li Y, Wang R, Zhang S. Moderate deviations for a stochastic heat equation with spatially correlated noise. Acta Appl Math, 2015, 139(1): 59–80

[26] Liu W. Large deviations for stochastic evolution equations with small multiplicative noise. Appl Math Optim, 2010, 61(1): 27–56

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

[28] Mao X, Matina J. Khasminskii-type theorems for stochastic differential delay equations. Stoch Anal Appl, 2005, 23(5): 1045–1069

[29] Maroulas V. Uniform large deviations for infinite dimensional stochastic systems with jumps. Mathematika, 2011, 57(1): 175–192

[30] Mohammed S E A. Stochastic Functional Differential Equations.Boston: Longman Scientific & Technical, 1984

[31] Ren J, Zhang X. Large deviations for homeomorphism ows of non-Lipschitz SDE. Bull Sci Math, 2005, 129(8): 643–655

[32] Sritharan S, Sundar P. Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise. Stochastic Process Appl, 2006, 116(11): 1636–1659

[33] Wang R, Zhai J, Zhang T. A moderate deviation principle for 2-D stochastic Navier-Stokes equations. J Differential Equations, 2015, 258(10): 3363–3390

[34] Wang R, Zhang T. Moderate deviations for stochastic reaction-diffusion equations with multiplicative noise. Potential Anal, 2015, 42(1): 99–113

[35] Xiong J. Large deviations for diffusion processes in duals of nuclear spaces. Appl Math Optim, 1996, 34(1): 1–27

[36] Zhai J, Zhang T. Large deviations for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises. Bernoulli, 2015, 21(4): 2351–2392

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