Taiwanese Journal of Mathematics


Tien Dung Nguyen

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In this paper we prove the variation of parameters formula for linear Volterra integro-differential equations driven by multifractional Brownian motion. To do this, an approximate result for the Stratonovich stochastic integral with respect to the multifractional Brownian motion is given. Based on our obtained results we study almost surely exponentially convergence of the solution. Also, the existence and uniqueness of the solution of a multifractional Volterra integro-differential equation with time delay are proved.

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Taiwanese J. Math., Volume 17, Number 1 (2013), 333-350.

First available in Project Euclid: 10 July 2017

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Zentralblatt MATH identifier

Primary: 45D05: Volterra integral equations [See also 34A12] 60G22: Fractional processes, including fractional Brownian motion 60H07: Stochastic calculus of variations and the Malliavin calculus

Volterra integro-differential equations variation of parameters formula multifractional Brownian motion Malliavin calculus


Nguyen, Tien Dung. LINEAR MULTIFRACTIONAL STOCHASTIC VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS. Taiwanese J. Math. 17 (2013), no. 1, 333--350. doi:10.11650/tjm.17.2013.1728. https://projecteuclid.org/euclid.twjm/1499705891

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  • E. Alòs, O. Mazet and D. Nualart, Stochastic Calculus with Respect to Gaussian Processes, The Annals of Probability, 29(2) (2001), 766-801.
  • J. A. D. Appleby and A. Freeman, Exponential asymptotic stability of linear Itô-Volterra equations with damped stochastic perturbations, Electron. J. Probab., 8(22) (2003), 22.
  • P. R. Bertrand, A. Hamdouni and S. Khadhraoui, Modelling NASDAQ Series by Sparse Multifractional Brownian Motion, Methodology and Computing in Applied Probability, (23 July 2010), pp. 1-18.
  • T. A. Burton, Volterra integral and differential equations, Mathematics in Science and Engineering, 2nd ed., 202, Elsevier B. V., Amsterdam, 2005.
  • B. Boufoussi, M. Dozzi and R. Marty, Local time and Tanaka formula for a Volterra-type multifractional Gaussian process, Bernoulli, 16(4) (2010), 1294-1311.
  • H. E. Gollwitzer, A note on a functional inequality, Proc. Amer. Math. Soc., 23 (1969), 642-647.
  • M. Li, S. C. Lim, B. J. Hu and H. Feng, Towards describing multi-fractality of traffic using local Hurst function, in: Lecture Notes in Computer Science, Vol. 4488, Springer, 2007, pp. 1012-1020.
  • S. C. Lim, Fractional Brownian motion and multifractional Brownian motion of Riemann-Liouville type, J. Phys. A: Math. Gen., 34 (2001), 1301-1310.
  • X. Mao, Exponential Stability of Stochastic Differential Equations, Vol. 182 of Pure and Applied Mathematics, Marcel Dekker, New York, 1994.
  • S. Murakami, Exponential asymptotic stability for scalar linear Volterra equations, Differential Integral Equations, 4(2) (1991), 519-525.
  • D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Springer, 2006.
  • N. Privault, Skorohod stochastic integration with respect to non-adapted processes on Wiener space, Stochastics Stochastics Rep., 65(1-2) (1998), 13-39.