# American Institute of Mathematical Sciences

February  2020, 25(2): 507-528. doi: 10.3934/dcdsb.2019251

## Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process

 1 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, Spain 2 Université d'Abomey-Calavi(UAC), Institut de Mathématiques et de Sciences Physiques(IMSP), 01 B.P. 613, Porto-Novo, République du Bénin 3 Université Gaston Berger de Saint-Louis, UFR SAT Département de Mathématiques, B.P234, Saint-Louis, Sénégal

Dedicated to Prof. Dr. Juan J. Nieto on the occasion of his 60th birthday

Received  January 2019 Revised  March 2019 Published  November 2019

Fund Project: This work has been partially supported by FEDER and the Spanish Ministerio de Economía y Competitividad project MTM2015-63723-P and the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under Proyecto de Excelencia P12-FQM-1492

The existence and uniqueness of mild solution of an impulsive stochastic system driven by a Rosenblatt process is analyzed in this work by using the Banach fixed point theorem and the theory of resolvent operator developed by R. Grimmer in [12]. Furthermore, the exponential stability in mean square for the mild solution to neutral stochastic integro–differential equations with Rosenblatt process is obtained by establishing an integral inequality. Finally, an example is exhibited to illustrate the abstract theory.

Citation: Tomás Caraballo, Carlos Ogouyandjou, Fulbert Kuessi Allognissode, Mamadou Abdoul Diop. Existence and exponential stability for neutral stochastic integro–differential equations with impulses driven by a Rosenblatt process. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 507-528. doi: 10.3934/dcdsb.2019251
##### References:
 [1] E. Alos, O. Maze and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29 (2001), 766-801.  doi: 10.1214/aop/1008956692.  Google Scholar [2] G. Arthi, J. H. Park and H. Y. Jung, Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion, Communications in Nonlinear Science and Numerical Simulations, 32 (2016), 145-157.  doi: 10.1016/j.cnsns.2015.08.014.  Google Scholar [3] P. Balasubramaniam, M. Syed Ali and J. H. Kim, Faedo-Galerkin approximate solutions for stochastic semilinear integrodifferential equations, Computers and Mathematics with Applications, 58 (2009), 48-57.  doi: 10.1016/j.camwa.2009.03.084.  Google Scholar [4] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis, 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.  Google Scholar [5] H. B. Chen, Integral inequality and exponential stability for neutral stochastic partial differential equations with delays, Journal of Inequalities and Applications, 2009 (2009), Art. ID 297478, 15 pp. doi: 10.1155/2009/297478.  Google Scholar [6] H. B. Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statistics and Probability Letters, 80 (2010), 50-56.  doi: 10.1016/j.spl.2009.09.011.  Google Scholar [7] H. B. Chen, The asymptotic behavior for second-order neutral stochastic partial differential equations with infinite delay, Discrete Dynamics in Nature and Society, 2011 (2011), Art. ID 584510, 15 pp. doi: 10.1155/2011/584510.  Google Scholar [8] J. Cui, L. T. Yan and X. C. Sun, Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps, Statistics and Probability Letters, 81 (2011), 1970-1977.  doi: 10.1016/j.spl.2011.08.010.  Google Scholar [9] M. Dieye, M. A. Diop and K. Ezzinbi, On exponential stability of mild solutions for some stochastic partial integrodifferential equations, Statistics Probability Letters, 123 (2017), 61-76.  doi: 10.1016/j.spl.2016.10.031.  Google Scholar [10] R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gaussian fields, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 50 (1979), 27-52.  doi: 10.1007/BF00535673.  Google Scholar [11] N. T. Dung, Stochastic Volterra integro-differential equations driven by fractional Brownian motion in a Hilbert space, Stochastics, 87 (2015), 142-159.  doi: 10.1080/17442508.2014.924938.  Google Scholar [12] R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349.  doi: 10.1090/S0002-9947-1982-0664046-4.  Google Scholar [13] F. Jiang and Y. Shen, Stability of impulsive stochastic neutral partial differential equations with infinite delays, Asian Journal of Control, 14 (2012), 1706-1709.  doi: 10.1002/asjc.491.  Google Scholar [14] A. N. Kolmogorov, The Wiener spiral and some other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR, 26 (1940), 115-118.   Google Scholar [15] I. Kruk, F. Russo and C. A. Tudor, Wiener integrals, Malliavin calculus and covariance measure structure, Journal of Functional Analysis, 249 (2007), 92-142.  doi: 10.1016/j.jfa.2007.03.031.  Google Scholar [16] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. doi: 10.1142/0906.  Google Scholar [17] N. N. Leonenko and V. V. Ahn, Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence, Journal of Applied Mathematics and Stochastic Analysis, 14 (2001), 27-46.  doi: 10.1155/S1048953301000041.  Google Scholar [18] J. Liang, J. H. Liu and T.-J. Xiao, Nonlocal problems for integrodifferential equations, Dynamics of Continuous, Discrete and Impulsive Systems, Series A, Mathematical Analysis, 15 (2008), 815-824.   Google Scholar [19] M. Maejima and C. A. Tudor, Wiener integrals with respect to the Hermite process and a non central limit theorem, Stochastic Analysis and Applications, 25 (2007), 1043-1056.  doi: 10.1080/07362990701540519.  Google Scholar [20] M. Maejima and C. A. Tudor, Selfsimilar processes with stationary increments in the second Wiener chaos, Probability and Mathematical Statistics, 32 (2012), 167-186.   Google Scholar [21] M. Maejima and C. A. Tudor, On the distribution of the Rosenblatt process, Statistics and Probability Letters, 83 (2013), 1490-1495.  doi: 10.1016/j.spl.2013.02.019.  Google Scholar [22] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-437.  doi: 10.1137/1010093.  Google Scholar [23] V. Pipiras and M. S. Taqqu, Regularization and integral representations of Hermite processes, Statistics and Probability Letters, 80 (2010), 2014-2023.  doi: 10.1016/j.spl.2010.09.008.  Google Scholar [24] J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar [25] Y. Ren, W. S. Yin and R. Sakthivel, Stabilization of stochastic differential equations driven by $G$-Brownian motion with feedback control based on discrete-time state observation, Automatica J. IFAC, 95 (2018), 146-151.  doi: 10.1016/j.automatica.2018.05.039.  Google Scholar [26] R. Sakthivel, P. Revathi, Y. Ren and G. J. Shen, Retarded stochastic differential equations with infinite delay driven by Rosenblatt process, Stochastic Analysis and Applications, 36 (2018), 304-323.  doi: 10.1080/07362994.2017.1399801.  Google Scholar [27] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 14. World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664.  Google Scholar [28] M. Ali Syed, Robust stability of stochastic fuzzy impulsive recurrent neural networks with time-varying delays, Iranian Journal of Fuzzy Systems, 11 (2014), 1–13, 97.  Google Scholar [29] M. S. Taqqu, Weak convergence to the fractional Brownian motion and to the Rosenblatt process, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31 (1974/75), 287-302.  doi: 10.1007/BF00532868.  Google Scholar [30] S. Tindel, C. A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probability Theory and Related Fields, 127 (2003), 186-204.  doi: 10.1007/s00440-003-0282-2.  Google Scholar [31] C. A. Tudor, Analysis of the Rosenblatt process, ESAIM Probab. Stat., 12 (2008), 230-257.  doi: 10.1051/ps:2007037.  Google Scholar [32] L. T. Yan and G. J. Shen, On the collision local time of sub-fractional Brownian motions, Stat. Probab. Lett., 80 (2010), 296-308.  doi: 10.1016/j.spl.2009.11.003.  Google Scholar [33] H. Yang and F. Jiang, Exponential stability of mild solutions to impulsive stochastic neutral partial differential equations with memory, Advances in Difference Equations, 2013 (2013), Art. ID 148, 9 pp. doi: 10.1186/1687-1847-2013-148.  Google Scholar

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##### References:
 [1] E. Alos, O. Maze and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29 (2001), 766-801.  doi: 10.1214/aop/1008956692.  Google Scholar [2] G. Arthi, J. H. Park and H. Y. Jung, Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion, Communications in Nonlinear Science and Numerical Simulations, 32 (2016), 145-157.  doi: 10.1016/j.cnsns.2015.08.014.  Google Scholar [3] P. Balasubramaniam, M. Syed Ali and J. H. Kim, Faedo-Galerkin approximate solutions for stochastic semilinear integrodifferential equations, Computers and Mathematics with Applications, 58 (2009), 48-57.  doi: 10.1016/j.camwa.2009.03.084.  Google Scholar [4] T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis, 74 (2011), 3671-3684.  doi: 10.1016/j.na.2011.02.047.  Google Scholar [5] H. B. Chen, Integral inequality and exponential stability for neutral stochastic partial differential equations with delays, Journal of Inequalities and Applications, 2009 (2009), Art. ID 297478, 15 pp. doi: 10.1155/2009/297478.  Google Scholar [6] H. B. Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statistics and Probability Letters, 80 (2010), 50-56.  doi: 10.1016/j.spl.2009.09.011.  Google Scholar [7] H. B. Chen, The asymptotic behavior for second-order neutral stochastic partial differential equations with infinite delay, Discrete Dynamics in Nature and Society, 2011 (2011), Art. ID 584510, 15 pp. doi: 10.1155/2011/584510.  Google Scholar [8] J. Cui, L. T. Yan and X. C. Sun, Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps, Statistics and Probability Letters, 81 (2011), 1970-1977.  doi: 10.1016/j.spl.2011.08.010.  Google Scholar [9] M. Dieye, M. A. Diop and K. Ezzinbi, On exponential stability of mild solutions for some stochastic partial integrodifferential equations, Statistics Probability Letters, 123 (2017), 61-76.  doi: 10.1016/j.spl.2016.10.031.  Google Scholar [10] R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gaussian fields, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 50 (1979), 27-52.  doi: 10.1007/BF00535673.  Google Scholar [11] N. T. Dung, Stochastic Volterra integro-differential equations driven by fractional Brownian motion in a Hilbert space, Stochastics, 87 (2015), 142-159.  doi: 10.1080/17442508.2014.924938.  Google Scholar [12] R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349.  doi: 10.1090/S0002-9947-1982-0664046-4.  Google Scholar [13] F. Jiang and Y. Shen, Stability of impulsive stochastic neutral partial differential equations with infinite delays, Asian Journal of Control, 14 (2012), 1706-1709.  doi: 10.1002/asjc.491.  Google Scholar [14] A. N. Kolmogorov, The Wiener spiral and some other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR, 26 (1940), 115-118.   Google Scholar [15] I. Kruk, F. Russo and C. A. Tudor, Wiener integrals, Malliavin calculus and covariance measure structure, Journal of Functional Analysis, 249 (2007), 92-142.  doi: 10.1016/j.jfa.2007.03.031.  Google Scholar [16] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. doi: 10.1142/0906.  Google Scholar [17] N. N. Leonenko and V. V. Ahn, Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence, Journal of Applied Mathematics and Stochastic Analysis, 14 (2001), 27-46.  doi: 10.1155/S1048953301000041.  Google Scholar [18] J. Liang, J. H. Liu and T.-J. Xiao, Nonlocal problems for integrodifferential equations, Dynamics of Continuous, Discrete and Impulsive Systems, Series A, Mathematical Analysis, 15 (2008), 815-824.   Google Scholar [19] M. Maejima and C. A. Tudor, Wiener integrals with respect to the Hermite process and a non central limit theorem, Stochastic Analysis and Applications, 25 (2007), 1043-1056.  doi: 10.1080/07362990701540519.  Google Scholar [20] M. Maejima and C. A. Tudor, Selfsimilar processes with stationary increments in the second Wiener chaos, Probability and Mathematical Statistics, 32 (2012), 167-186.   Google Scholar [21] M. Maejima and C. A. Tudor, On the distribution of the Rosenblatt process, Statistics and Probability Letters, 83 (2013), 1490-1495.  doi: 10.1016/j.spl.2013.02.019.  Google Scholar [22] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-437.  doi: 10.1137/1010093.  Google Scholar [23] V. Pipiras and M. S. Taqqu, Regularization and integral representations of Hermite processes, Statistics and Probability Letters, 80 (2010), 2014-2023.  doi: 10.1016/j.spl.2010.09.008.  Google Scholar [24] J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar [25] Y. Ren, W. S. Yin and R. Sakthivel, Stabilization of stochastic differential equations driven by $G$-Brownian motion with feedback control based on discrete-time state observation, Automatica J. IFAC, 95 (2018), 146-151.  doi: 10.1016/j.automatica.2018.05.039.  Google Scholar [26] R. Sakthivel, P. Revathi, Y. Ren and G. J. Shen, Retarded stochastic differential equations with infinite delay driven by Rosenblatt process, Stochastic Analysis and Applications, 36 (2018), 304-323.  doi: 10.1080/07362994.2017.1399801.  Google Scholar [27] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 14. World Scientific Publishing Co., Inc., River Edge, NJ, 1995. doi: 10.1142/9789812798664.  Google Scholar [28] M. Ali Syed, Robust stability of stochastic fuzzy impulsive recurrent neural networks with time-varying delays, Iranian Journal of Fuzzy Systems, 11 (2014), 1–13, 97.  Google Scholar [29] M. S. Taqqu, Weak convergence to the fractional Brownian motion and to the Rosenblatt process, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31 (1974/75), 287-302.  doi: 10.1007/BF00532868.  Google Scholar [30] S. Tindel, C. A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion, Probability Theory and Related Fields, 127 (2003), 186-204.  doi: 10.1007/s00440-003-0282-2.  Google Scholar [31] C. A. Tudor, Analysis of the Rosenblatt process, ESAIM Probab. Stat., 12 (2008), 230-257.  doi: 10.1051/ps:2007037.  Google Scholar [32] L. T. Yan and G. J. Shen, On the collision local time of sub-fractional Brownian motions, Stat. Probab. Lett., 80 (2010), 296-308.  doi: 10.1016/j.spl.2009.11.003.  Google Scholar [33] H. Yang and F. Jiang, Exponential stability of mild solutions to impulsive stochastic neutral partial differential equations with memory, Advances in Difference Equations, 2013 (2013), Art. ID 148, 9 pp. doi: 10.1186/1687-1847-2013-148.  Google Scholar
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