# 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  February 2020 Early access  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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] A. N. Kolmogorov, The Wiener spiral and some other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR, 26 (1940), 115-118. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [31] C. A. Tudor, Analysis of the Rosenblatt process, ESAIM Probab. Stat., 12 (2008), 230-257.  doi: 10.1051/ps:2007037. [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. [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.

show all references

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

##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] A. N. Kolmogorov, The Wiener spiral and some other interesting curves in Hilbert space, Dokl. Akad. Nauk SSSR, 26 (1940), 115-118. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [31] C. A. Tudor, Analysis of the Rosenblatt process, ESAIM Probab. Stat., 12 (2008), 230-257.  doi: 10.1051/ps:2007037. [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. [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.
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