TIME-DEPENDENT NEUTRAL STOCHASTIC DELAY PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY ROSENBLATT PROCESS IN HILBERT SPACE

Abstract

In this paper, we investigate a class of time-dependent neutral stochastic functional dierential equations with nite delay driven by Rosenblatt process in a real separable Hilbert space. We prove the existence of unique mild solution by the well-known Banach xed point principle. At the end we provide a practical example in order to illustrate the viability of our result.

Keywords

• Neutral stochastic evolution equations,Evolution operator,Rosenblatt process,Wiener integral,Banach fixed point theorem

References

1. P. Acquistapace and B. Terreni., A unified approach to abstract linear parabolic equations, Tend. Sem. Mat. Univ. Padova, Vol. 78, pp. 47-107 (1987).
2. D. Aoued and S. Baghli, Mild solutions for Perturbed evolution equations with infinite state-dependent delay, Electronic Journal of Qualitative Theory of Differential Equations, Vol. 59, pp. 1-24 (2013).
3. B. Boufoussi, S. Hajji and E. Lakhel, Functional differential equations in Hilbert spaces driven by a fractional Brownian motion, Afrika Matematika, Vol. 23, N. 2, pp. 173-194 (2011).
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, Vol. 74, pp. 3671-3684 (2011).
5. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, Cambridge University Press, Cambridge, (1992).
6. S. Hajji and E. Lakhel, Existence and uniqueness of mild solutions to neutral SFDEs driven by a fractional Brownian motion with non-Lipschitz coecients, Journal of Numerical Mathematics and Stochastics, Vol. 7, N. 1, pp. 14-29 (2015).
7. Lakhel, E. Exponential stability for stochastic neutral functional differential equations driven by Rosenblatt process with delay and Poisson jumps, Random Oper. Stoch. Equ., Vol. 24, N. 2, pp. 113-127 (2016).
8. N. N. Leonenko and V.V. Ahn., Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence, J. Appl. Math. Stoch. Anal., Vol. 14, pp. 27-46 (2001).

9. M. Maejima and C. A. Tudor, On the distribution of the Rosenblatt process, Statist. Probab. Letters, Vol. 83, pp. 1490-1495 (2013).
10. M. Maejima, C. A. Tudor, Wiener integrals with respect to the Hermite process and a non central limit theorem, Stoch. Anal. Appl., Vol. 25, pp. 1043-1056 (2007).
11. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, (1983).
12. V. Pipiras and M.S. Taqqu, Integration questions related to the fractional Brownian motion. Probability Theory and Related Fields, Vol. 118, pp. 251-281 (2001).
13. M. Rosenblatt, Independence and dependence, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Contributions to Probability Theory, pp. 431-443, University of California Press, Berkeley, Calif., (1961).
14. M. S. Taqqu, Weak convergence to fractional Brownian motion and the Rosenblatt Process, Z. Wahr. Geb., Vol. 31, pp. 287-302 (1975).
15. M. Taqqu, Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrschein-lichkeitstheor, Verw. Geb., Vol. 50, pp. 53-83 (1979).
16. C. A. Tudor, Analysis of the Rosenblatt process, ESAIM Probab. Statist., Vol. 12, pp. 230-257 (2008).