Abstract
This paper is devoted to derive a Freidlin–Wentzell type of the large deviation principle for stochastic differential equations with general delayed generator. We improve the result of Chi Mo and Jiaowan Luo [C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal. 80 2013, 202–210].
Acknowledgements
This work was done during the stay of the first author (Clément Manga) at UFR Mathématiques et Informatique of Université Félix H. Boigny, Cocody (Côte d’Ivoire). He would like to thank all its administration for their support and hospitality. We also thank the anonymous referee for their comments and suggestions which allowed us to improve the final version of this article.
References
[1] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab. 36 (2008), no. 4, 1390–1420. 10.21236/ADA476159Search in Google Scholar
[2] S. Cerrai and M. Röckner, Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term, Ann. Probab. 32 (2004), no. 1B, 1100–1139. 10.1016/j.anihpb.2004.03.001Search in Google Scholar
[3] L. U. Delong and P. Imkeller, Backward stochastic differential equations with time delayed generators—results and counterexamples, Ann. Appl. Probab. 20 (2010), no. 4, 1512–1536. 10.1214/09-AAP663Search in Google Scholar
[4] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Appl. Math. (N. Y.) 38, Springer, New York, 1997. 10.1007/978-1-4612-5320-4Search in Google Scholar
[5] M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. I, Comm. Pure Appl. Math. 28 (1975), 1–47. 10.1002/cpa.3160280102Search in Google Scholar
[6] M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. II, Comm. Pure Appl. Math. 28 (1975), 279–301. 10.1002/cpa.3160280206Search in Google Scholar
[7] M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. III, Comm. Pure Appl. Math. 29 (1976), no. 4, 389–461. 10.1002/cpa.3160290405Search in Google Scholar
[8] P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley Ser. Probab. Stat., John Wiley & Sons, New York, 1997. 10.1002/9781118165904Search in Google Scholar
[9] I. Elsanosi, B. Ø ksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics Stochastics Rep. 71 (2000), no. 1–2, 69–89. 10.1080/17442500008834259Search in Google Scholar
[10] M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer, New York, 1984. 10.1007/978-1-4684-0176-9Search in Google Scholar
[11] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional-differential Equations, Math. Appl. 463, Kluwer Academic, Dordrecht, 1999. 10.1007/978-94-017-1965-0Search in Google Scholar
[12] W. Liu, Large deviations for stochastic evolution equations with small multiplicative noise, Appl. Math. Optim. 61 (2010), no. 1, 27–56. 10.1007/s00245-009-9072-2Search in Google Scholar
[13] C. Mo and J. Luo, Large deviations for stochastic differential delay equations, Nonlinear Anal. 80 (2013), 202–210. 10.1016/j.na.2012.10.004Search in Google Scholar
[14] S.-E. A. Mohammed and T. Zhang, Large deviations for stochastic systems with memory, Discrete Contin. Dyn. Syst. Ser. B 6 (2006), no. 4, 881–893. 10.3934/dcdsb.2006.6.881Search in Google Scholar
[15] J. Ren and X. Zhang, Freidlin–Wentzell’s large deviations for homeomorphism flows of non-Lipschitz SDEs, Bull. Sci. Math. 129 (2005), no. 8, 643–655. 10.1016/j.bulsci.2004.12.005Search in Google Scholar
[16] M. Scheutzow, Qualitative behaviour of stochastic delay equations with a bounded memory, Stochastics 12 (1984), no. 1, 41–80. 10.1080/17442508408833294Search in Google Scholar
[17] S. S. Sritharan and P. Sundar, Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise, Stochastic Process. Appl. 116 (2006), no. 11, 1636–1659. 10.1016/j.spa.2006.04.001Search in Google Scholar
[18] D. W. Stroock, An Introduction to the Theory of Large Deviations, Universitext, Springer, New York, 1984. 10.1007/978-1-4613-8514-1Search in Google Scholar
[19] S. R. S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math. 19 (1966), 261–286. 10.1002/cpa.3160190303Search in Google Scholar
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