JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (12): 81-88.doi: 10.6040/j.issn.1671-9352.0.2017.089

Previous Articles     Next Articles

Existence and controllability of nonlocal stochastcic integro-differential equations driven by fractional Brownian motion

CUI Jing, LIANG Qiu-ju   

  1. School of Mathematics &
    Computer Science, Anhui Normal University, Wuhu 241000, Anhui, China
  • Received:2017-03-10 Online:2017-12-20 Published:2017-12-22

Abstract: We study the existence and controllability of nonlocal stochastcic integro-differential equations driven by fractional Brownian motion in a real separable Hilbert space. Sufficient conditions are derived by using the Banach fixed point theorem and stochastic analysis thoery. An example is provided to illustrate the theory.

Key words: controllability, fractional Brownian motion, nonlocal condition, stochastcic integro-differential equations

CLC Number: 

  • O211.63
[1] DUNCAN T E, PASIKDUNCAN B, MASLOWSKI B. Fractional Brownian motion and stochastic equations in Hilbert spaces[J]. Stochastics and Dynamics, 2011, 2(2):225-250.
[2] FEYEL D, PRADELLE A D L. On fractional Brownian processes[J]. Potential Analasis, 1999, 10(3):273-288.
[3] CHRISTODOULOU-VOLOS C, SIOKIS F M. Long range dependence in stock market returns[J]. Applied Financial Economics, 2006, 16(18):1331-1338.
[4] MASLOWSKI B, NUALART D. Evolution equations driven by a fractional Brownian motion[J]. Journal of Functional Analasis, 2003, 202(1):277-305.
[5] MILLER R K. An integro-differential equation for heat conductors with memory[J]. Journal of Mathematical Analasis and Applications, 1978, 66(2):313-332.
[6] LAKSMIKANTHAM V, RAMA M R M. Theory of integro-differential Equations[M]. London: Gordon and Breach Science, 1995: 275-283.
[7] LEVIN J J, NOHEL J A. The integro-differential equations of a class of nuclear reactors with delayed neutrons[J]. Archive for Rational Mechanics Analysis, 1968, 31(2):151-172.
[8] DEBBOUCHE A, BALEANU D. Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems[J]. Computers Mathematics with Applications, 2015, 62(3):1442-1450.
[9] BALACHANDRANK K, DIVYA S, RIVERO M, et al. Controllability of nonlinear implicit neutral fractional volterra integrodifferential systems[J]. Journal of Vibration and Control, 2015, 21(9):641-646.
[10] AISSANI K, BENCHOHRA M. Controllability of fractional integrodifferential equations with state-dependent delay[J]. Integral Equations and Applications, 2015, 28(2):149-167.
[11] LAKHEL E H. Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion[J]. Stochastic Analysis and Applications, 2016, 34(3):427-440.
[12] CUI Jing, YAN Litan. Controllability of neutral stochastic evolution equations driven by fractional Brownian motion[J]. Acta Mathematica Scientia, 2017, 37B(1):1-12.
[13] BYSZEWSKI L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem[J]. Journal of Mathematical Analysis and Applications, 1991, 162(2):494-505.
[14] KENG Deng. Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions[J]. Journal of Mathematical Analysis and Applications, 1993, 179(2):630-637.
[15] DEBBOUCHE A. Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems[J]. Advances in Difference Equations, 2011, 2011(1):1-10.
[16] DEBBOUCHE A. Fractional evolution integro-differential systems with nonlocal conditions[J]. Advances in Dynamical Systems and Applications, 2010, 5(1):49-60.
[17] LI Fang. Existence and uniqueness of mild solution for fractional intergrodifferential equations of neutral type with nonlocal conditions[J]. Mathematica Slovaca, 2012, 62(5):921-936.
[18] CUI Jing, WANG Zhi. Nonlocal stochastic integro-differential equations driven by fractional Brownian motion[J]. Advances in Difference Equations, 2016, 2016(1):115.
[19] NUALART D. The Malliavin calculus and related topics[M]. Berlin: Springer-Verlag, 2006.
[20] GRIMMER R C. Resolvent operators for Integral equations in a Banach space[J]. Transactions of the American Mathematical Society, 1982, 273(1):333-349.
[21] KLAMKA J. Stochastic controllability of linear systems with delay in control[J]. Bulletin of the Polish Academy of Sciences Technical Sciences, 2007, 55(1):23-29.
[1] DENG Long-juan, ZHU Dong-jin, SHEN Guang-jun. Power variation of weighted-fractional Brownian motion and application [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(06): 19-26.
[2] DENG Lei, ZHAO Jian-li, LIU Hua, LI Ying. Controllability and observability of k-valued control networks [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(04): 27-35.
[3] YANG Zhao-qiang. A kind of European lookback option pricing model under fractional  jump-diffusion mixed fractional Brownian motion [J]. J4, 2013, 48(6): 67-74.
[4] SHEN Ming-xuan1, HE Chao-lin2. Geometric average asian option pricing in fractional brownian environment [J]. J4, 2013, 48(3): 48-52.
[5] YANG Zhao-qiang. A kind European lookback option pricing model in mixed fractional Brownian motion environment [J]. J4, 2012, 47(9): 105-109.
[6] SHEN Guang-jun 1,2, HE Kun 3, YAN Li-tan 3. Remarks on sub-fractional Brownian motion [J]. J4, 2011, 46(3): 102-108.
[7] YANG He. Existence of mild solutions for impulsive evolution equations with nonlocal conditions in the α-norm [J]. J4, 2011, 46(11): 70-74.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!