Existence and stability results for neutral stochastic delay differential equations driven by a fractional Brownian motion

  • Authors

    • R. Maheswari Sri Eshwar college of engineering
    • S. Karunanithi Kongunadu Arts and Science College
    2015-03-12
    https://doi.org/10.14419/ijamr.v4i2.4175
  • Asymptotic Behaviors, Delays, Fractional Brownian Motion, Mild Solution, Wiener Integral.

  • Abstract

    In this paper we investigate the existence, uniqueness, asymptotic behavior of mild solutions to neutral stochastic differential equations with delays driven by a fractional Brownian motion in a Hilbert space. The cases of finite and infinite delays are analyzed.

  • References

    1. [1] L. Coutin, Z. Qian, Stochastic differential equations for fractional Brownian motions. ComptesRendus de I'Académie des Sciences-Series I-Mathematics 331 (2000) 75-80. http://dx.doi.org/10.1016/S0764-4442(00)01594-9.

      [2] Y. Hu, D. Nualart, Differential equations driven by H¨older continuous functions of order greater than 1/2, Stochastic analysis and applications 2 (2007) 399-413. http://dx.doi.org/10.1007/978-3-540-70847-6_17.

      [3] D. Nualart, Y. Ouknine, Regularization of differential equations by fractional noise, Stochastic Process and their Applications 102 (2002) 103-116. http://dx.doi.org/10.1016/S0304-4149(02)00155-2.

      [4] D. Nualart, A. R˘a¸scanu, Differential equations driven by fractional Brownian motion, CollectaneaMathematica 53 (2002) 55-81.

      [5] D. Nualart, B. Saussereau, Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stochastic Processes and their Applications119 (2009) 391-409. http://dx.doi.org/10.1016/j.spa.2008.02.016.

      [6] T. Caraballo, J. Real, T. Taniguchi, The exponential stability of neutral stochastic delay partial differential equations, Discrete and Continuous Dynamical Systems 18 (2007) 295–313. http://dx.doi.org/10.3934/dcds.2007.18.295.

      [7] W. Dai, C. C. Heyde, Ito's formula with respect to fractional Brownian motion and its application, Journal of Applied Mathematics and Stochastic Analysis 9 (1996) 439–448. http://dx.doi.org/10.1155/S104895339600038X.

      [8] D. Feyel, A. de la Pradelle, on fractional Brownian processes, Potential Analysis 10 (1999) 273–288. http://dx.doi.org/10.1023/A:1008630211913.

      [9] E. Hernandez, D. N. Keck, M. A. McKibben, On a class of measure-dependent stochastic evolution equations driven by fBm, Journal of Applied Mathematics and Stochastic Analysis 2007 (2007) 1-26. http://dx.doi.org/10.1155/2007/69747.

      [10] P. H. Bezandry, Existence of almost periodic solutions for semilinear stochastic evolution equations driven by fractional Brownian motion, Electronic Journal of Differential Equations 2012 (2012) 1–21.

      [11] T. E. Duncan, B. Maslowski, B. Pasik-Duncan, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stochastics and Dynamics 2 (2002) 225–250. http://dx.doi.org/10.1142/S0219493702000340.

      [12] B. Maslowski, D. Nualart, Evolution equations driven by a fractional Brownian motion, Journal of Functional Analysis 2 (2003) 277–305. http://dx.doi.org/10.1016/S0022-1236(02)00065-4.

      [13] T. Caraballo, M. J. Garrido-Atienza, T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Analysis: Theory, Methods and Applications 74 (2011) 3671–3684. http://dx.doi.org/10.1016/j.na.2011.02.047.

      [14] B. Boufoussi, S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statistics & Probability Letters 82 (2012)1549–1558. http://dx.doi.org/10.1016/j.spl.2012.04.013.

      [15] T. E. Duncan, B. Maslowski, B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochastic process and their applications 115(8) (2005) 1357-1383.

      [16] Nguyen-Tien-Dung, Neutral stochastic differential equations driven by a fractional Brownian motion with impulsive effects and varying time delays, Journal of the Korean Statistical society 43 (2014) 599–608. http://dx.doi.org/10.1016/j.jkss.2014.02.003.

      [17] T. A. Burton. Stability by fixed point theory for functional differential equations, Mineola, NY, Dover Publications, Inc. (2006).

      [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York. (1983). http://dx.doi.org/10.1007/978-1-4612-5561-1.

      [19] G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, (1992). http://dx.doi.org/10.1017/CBO9780511666223.

  • Downloads

  • How to Cite

    Maheswari, R., & Karunanithi, S. (2015). Existence and stability results for neutral stochastic delay differential equations driven by a fractional Brownian motion. International Journal of Applied Mathematical Research, 4(2), 281-294. https://doi.org/10.14419/ijamr.v4i2.4175

    Received date: 2015-01-13

    Accepted date: 2015-02-10

    Published date: 2015-03-12