Fractional neutral stochastic differential equations driven by \(\alpha\)-stable process
- School of Information and Mathematics, Yangtze University, Jingzhou 434023, China.
In this paper, we are concerned with a class of fractional neutral stochastic partial differential equations driven by \(\alpha\)-stable process. By the stochastic analysis technique, the properties of operator semigroup and combining the Banach fixed-point theorem, we prove the existence and uniqueness of the mild solutions to this kind of equations driven by \(\alpha\)-stable process. In the end, an example is given to demonstrate the theory of our work.
Share and Cite
Zhi Li, Fractional neutral stochastic differential equations driven by \(\alpha\)-stable process, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4713--4723
Li Zhi, Fractional neutral stochastic differential equations driven by \(\alpha\)-stable process. J. Nonlinear Sci. Appl. (2017); 10(9):4713--4723
Li, Zhi. "Fractional neutral stochastic differential equations driven by \(\alpha\)-stable process." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4713--4723
- Fractional neutral SDEs
- \(\alpha\)-stable process
- existence and uniqueness.
D. Applebaum, Lévy processes and stochastic calculus, Second edition, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2009)
D. Baleanu, H. Jafari, H. Khan, S. J. Johnston , Results for mild solution of fractional coupled hybrid boundary value problems, Open Math., 13 (2015), 601–608.
J.-H. Bao, C.-G. Yuan, Numerical analysis for neutral SPDEs driven by \(\alpha\)-stable processes, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014), 17 pages.
J. Cui, L.-T. Yan , Existence result for fractional neutral stochastic integro-differential equations with infinite delay, J. Phys. A, 44 (2011), 16 pages.
G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (1992)
Z. Dong, L.-H. Xu, X.-C. Zhang, Invariant measures of stochastic 2D Navier-Stokes equation driven by \(\alpha\)-stable processes , Electron. Commun. Probab., 16 (2011), 678–688.
U. Haagerup , The best constants in the Khintchine inequality, Studia Math., 70 (1982), 231–283.
R. Jahanipur , Nonlinear functional differential equations of monotone-type in Hilbert spaces, Nonlinear Anal., 72 (2010), 1393–1408.
F. Jiao, Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62 (2011), 1181–1199.
P. Kalamani, D. Baleanu, S. Selvarasu, M. M. Arjunan, On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions, Adv. Difference Equ., 163 (2016), 36 pages.
M. Kerboua, A. Debbouche, D. Baleanu , Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 16 pages.
K.-X. Li, Stochastic delay fractional evolution equations driven by fractional Brownian motion, Math. Methods Appl. Sci., 38 (2015), 1582–1591.
K.-X. Li, J.-G. Peng , Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys., 65 (2014), 941–959.
K.-X. Li, J.-G. Peng, J.-H. Gao, Existence results for semilinear fractional differential equations via Kuratowski measure of noncompactness, Fract. Calc. Appl. Anal., 15 (2012), 591–610.
K. Liu , Stability of infinite dimensional stochastic differential equations with applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL (2006)
C. Lizama, G. M. N'Guérékata, Mild solutions for abstract fractional differential equations, Appl. Anal., 92 (2013), 1731–1754.
J.-W. Luo, K. Liu, Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stochastic Process. Appl., 118 (2008), 864–895.
F. Mainardi, Fractional calculus and waves in linear viscoelasticity, An introduction to mathematical models, Imperial College Press, London (2010)
A. Pazy , Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Springer-Verlag, New York (1992)
E. Priola, J. Zabczyk , Structural properties of semilinear SPDEs driven by cylindrical stable processes, Probab. Theory Related Fields, 149 (2011), 97–137.
R. Sakthivel, P. Revathi, Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations, Nonlinear Anal., 81 (2013), 70–86.
R. Sakthivel, S. Suganya, S. M. Anthoni , Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl., 63 (2012), 660–668.
G. Samorodnitsky, M. S. Taqqu , Stable non-Gaussian random processes, Stochastic models with infinite variance, Stochastic Modeling, Chapman & Hall, New York (1994)
K.-I. Sato, Lévy processes and infinitely divisible distributions , Translated from the 1990 Japanese original, Revised by the author, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1999)
L.-H. Xu , Ergodicity of the stochastic real Ginzburg-Landau equation driven by \(\alpha\)-stable noises, Stochastic Process. Appl., 123 (2013), 3710–3736.
Y.-C. Zang, J.-P. Li, Stability in distribution of neutral stochastic partial differential delay equations driven by \(\alpha\)-stable process, Adv. Difference Equ., 2014 (2014), 16 pages.
X.-C. Zhang , Derivative formulas and gradient estimates for SDEs driven by \(\alpha\)-stable processes, Stochastic Process. Appl., 123 (2013), 1213–1228.
Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063–1077.
Y. Zhou, F. Jiao , Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. Real World Appl., 11 (2010), 4465–4475.