Neutral stochastic functional differential evolution equations driven by Rosenblatt process with varying-time delays
Keywords:Neutral stochastic evolution equations, Evolution operator, Rosenblatt process, Wiener integral, Banach fixed point theorem
AbstractHermite processes are self-similar processes with stationary increments, the Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by Rosenblatt process with index H ∈ (1/2, 1) which is a special case of a self-similar process with long-range dependence. More precisely, we prove the existence and uniqueness of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.
P. Acquistapace and B. Terreni. “A unified approach to abstract linear parabolic equations”, Rendiconti del seminario matematico della Università di Padova, vol. 78, pp. 47-107, 1987. [On line]. Available: https://bit.ly/2n1nvOT
D. Aoued and S. Baghli-Bendimerad, “Mild solutions for perturbed evolution equations with infinite state-dependent delay”, Electronic journal of qualitative theory of differential equations, no. 59, pp. 1–24, Oct. 2013, doi:10.14232/ejqtde.2013.1.59.
B. Boufoussi and S. Hajji, “Neutral stochastic functional differential equation driven by a fractional Brownian motion in a Hilbert space”, Statistics & probability letters, vol. 82 no. 8, pp. 1549-1558, Aug. 2012, doi: 10.1016/j.spl.2012.04.013.
B. Boufoussi, S. Hajji, and E. Lakhel, “Functional differential equations in Hilbert spaces driven by a fractional Brownian motion”, Afrika Matematika, vol. 23, no. 2, pp. 173–194, Jun. 2011, doi: 10.1007/s13370-011-0028-8.
G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Cambridge: Cambridge University Press, 1992. doi: 10.1017/CBO9780511666223.
E. Lakhel and S. Hajji. “Existence and uniqueness of mild solutions to neutral SFDEs driven by a fractional Brownian motion with nonLipschitz coefficients”, Journal of numerical mathematics and stochastics, vol.7, no. 1, pp. 14-29, 2015. [On line]. Available: https://bit.ly/2nSMKDe
A. Boudaoui and E. Lakhel, “Controllability of Stochastic Impulsive Neutral Functional Differential Equations Driven by Fractional Brownian Motion with Infinite Delay”, Differential Equations and Dynamical Systems, vol. 26, no. 1-3, pp. 247–263, Nov. 2017, doi: 10.1007/s12591-017-0401-7.
E. Lakhel and A. Tlidi, “Controllability of time-dependent neutral stochastic functional differential equations driven by a fractional Brownian motion”, Journal of nonlinear sciences and applications, vol. 11, no. 06, pp. 850–863, May 2018., doi: 10.22436/jnsa.011.06.11.
B. Boufoussi, S. Hajji, and E. Lakhel, “Exponential stability of impulsive neutral stochastic functional differential equation driven by fractional Brownian motion and Poisson point processes”, Afrika Matematika, vol. 29, no. 1-2, pp. 233–247, Nov. 2017., doi: 10.1007/s13370-017-0538-0.
E. Lakhel, “Controllability of neutral functional differential equations driven by fractional Brownian motion with infinite delay”, Nonlinear dynamics and systems theory, vol. 17, no. 3, pp. 291-302, 2017.
E. Lakhel and M. Mckibben, “Existence of solutions for fractional neutral functional differential equations driven by fBm with infinite delay”, Stochastics, vol. 90, no. 3, pp. 313–329, Jun. 2017, doi: 10.1080/17442508.2017.1346657.
N. Leonenko and V. Anh, “Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence”, Journal of applied mathematics and stochastic analysis, vol. 14, no. 1, pp. 27–46, Jan. 2001, doi: 10.1155/S1048953301000041.
M. Maejima and C. Tudor, “On the distribution of the Rosenblatt process”, Statistics & probability letters, vol. 83, no. 6, pp. 1490–1495, Jun. 2013, doi: 10.1016/j.spl.2013.02.019.
M. Maejima and C. Tudor, “Wiener integrals with respect to the Hermite process and a non-central limit theorem”, Stochastic analysis and applications, vol. 25, no. 5, pp. 1043–1056, Ago. 2007, doi: 10.1080/07362990701540519.
A. Pazy, Semigroups of linear operators and applications to partial differential equations, vol. 44. New York, NY: Springer, 1983, doi: 10.1007/978-1-4612-5561-1.
V. Pipiras and M. Taqqu, “Integration questions related to fractional Brownian motion”, Probability theory and related fields, vol. 118, no. 2, pp. 251–291, Oct. 2000, doi: 10.1007/s440-000-8016-7.
Y. Ren and D. Sun, “Second-order neutral impulsive stochastic evolution equations with delay”, Journal of mathematical physics, vol. 50, no. 10, p. 102709, Oct. 2009, doi: 10.1063/1.3251332.
R. Sakthivel, P. Revathi, Y. Ren, and G. Shen, “Retarded stochastic differential equations with infinite delay driven by Rosenblatt process”, Stochastic analysis and applications, vol. 36, no. 2, pp. 304–323, Dec. 2017., doi: 10.1080/07362994.2017.1399801.
R. Rosenblatt, “Independence and dependence”, in Proceedings of the Fourth Berkeley symposium on mathematical statistics and probability, 1961, vol. 2, pp. 431–443. [On line]. Available: https://bit.ly/2n6DkUw
M. Taqqu, “Weak convergence to fractional brownian motion and to the Rosenblatt process”, Zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete, vol. 31, no. 4, pp. 287–302, 1975, doi: 10.1007/BF00532868.
M. Taqqu, “Convergence of integrated processes of arbitrary Hermite rank”, Zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete, vol. 50, no. 1, pp. 53–83, 1979, doi: 10.1007/BF00535674.
C. Tudor, “Analysis of the Rosenblatt process”, ESAIM: Probability and statistics, vol. 12, pp. 230–257, Jan. 2008, doi: 10.1051/ps:2007037.
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