Abstract
We utilize the weak convergence method to establish the Freidlin–Wentzell large deviations principle (LDP) for stochastic delay differential equations (SDDEs) with super-linearly growing coefficients, which covers a large class of cases with non-globally Lipschitz coefficients. The key ingredient in our proof is the uniform moment estimate of the controlled equation, where we handle the super-linear growth of the coefficients by an iterative argument. Our results allow both the drift and diffusion coefficients of the considered equations to super-linearly grow not only with respect to the delay variable but also to the state variable. This work extends the existing results which develop the LDPs for SDDEs with super-linearly growing coefficients only with respect to the delay variable.
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References
Bao J., Yuan C., Large deviations for neutral functional SDEs with jumps. Stochastics, 2015, 87(1): 48–70
Boue M., Dupuis P., A variational representation for certain functionals of Brownian motion. Ann. Probab., 1998, 26(4): 1641–1659
Budhiraja A., Dupuis P., A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Statist., 2000, 20(1): 39–61
Budhiraja A., Dupuis P., Maroulas V., Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab., 2008, 36(4): 1390–1420
Chen C., Chen Z., Hong J., ** D., Large deviations principles of sample paths and invariant measures of numerical methods for parabolic SPDEs. 2021, ar**v:2106.11018
Chen C., Hong J., ** D., Sun L., Asymptotically-preserving large deviations principles by stochastic symplectic methods for a linear stochastic oscillator. SIAM J. Numer. Anal., 2021, 59(1): 32–59
Chen C., Hong J., ** D., Sun L., Large deviations principles for symplectic discretizations of stochastic linear Schrödinger equation. Potential Anal., 2023, 59(3): 971–1011
Chen X., Random Walk Intersections—Large Deviations and Related Topics. Math. Surveys Monogr., 157, Providence, RI: American Mathematical Society, 2010
Chiarini A., Fischer M., On large deviations for small noise Ito processes. Adv. in Appl. Probab., 2014, 46(4): 1126–1147
Dembo A., Zeitouni O., Large Deviations Techniques and Applications. Stoch. Model. Appl. Probab., 38, Berlin: Springer-Verlag, 2010
Dupuis P., Ellis R.S., A Weak Convergence Approach to the Theory of Large Deviations. New York: John Wiley & Sons, Inc., 1997
Freidlin M.I., Wentzell A.D., Random Perturbations of Dynamical Systems. New York: Springer-Verlag, 1984
Hong J., ** D., Sheng D., Numerical approximations of one-point large deviations rate functions of stochastic differential equations with small noise. 2021, ar**v:2102.04061
Hong J., ** D., Sheng D., Sun L., Numerically asymptotical preservation of the large deviations principles for invariant measures of Langevin equations. 2020, ar**v:2009.13336
Klenke A., Probability Theory—A Comprehensive Course. Universitext, London: Springer-Verlag London, Ltd., 2008
Mao X., Rassias M.J., Khasminskii-type theorems for stochastic differential delay equations. Stoch. Anal. Appl., 2005, 23(5): 1045–1069
Matoussi A., Sabbagh W., Zhang T., Large deviation principles of obstacle problems for quasilinear stochastic PDEs. Appl. Math. Optim., 2021, 83(2): 849–879
Mo C., Luo J., Large deviations for stochastic differential delay equations. Nonlinear Anal., 2013, 80: 202–210
Mohammed S.A., Zhang T., Large deviations for stochastic systems with memory. Discrete Contin. Dyn. Syst. Ser. B, 2006, 6(4): 881–893
Scheutzow M., Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics, 1984, 12(1): 41–80
Scheutzow M., A stochastic Gronwall lemma. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2013, 16(2): Paper No. 1350019, 4 pp.
Suo Y., Yuan C., Large deviations for neutral stochastic functional differential equations. Commun. Pure Appl. Anal., 2020, 19(4): 2369–2384
Acknowledgements
The authors would like to thank the editors for their help and the referees for their constructive and insightful comments. This work was supported by National Natural Science Foundation of China (Nos. 12201228, 12201552, 12031020, 11971470, 11971488), the National key R&D Program of China (No. 2020YFA0713701), the Fundamental Research Funds for the Central Universities (No. 3004011142), Yunnan Fundamental Research Projects (No. 202301AU070010) and Innovation Team of School of Mathematics and Statistics of Yunnan University (No. ST20210104).
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**, D., Chen, Z. & Zhou, T. Large Deviations Principle for Stochastic Delay Differential Equations with Super-linearly Growing Coefficients. Front. Math (2024). https://doi.org/10.1007/s11464-023-0072-3
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DOI: https://doi.org/10.1007/s11464-023-0072-3
Keywords
- Large deviations principle
- stochastic delay differential equations
- super-linear growth
- weak convergence method