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Stationarity of the Solution for the Semilinear Stochastic Integral Equation on the Whole Real Line

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Book cover Malliavin Calculus and Stochastic Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 34))

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Abstract

In this article we prove the stationarity of the solution of the H-valued integral equation

$$X(t) =\displaystyle\int _{ -\infty }^{t}U(t - s)f(X(s))\mathrm{d}s + V (t),$$

where H is a real separable Hilbert space. In this equation, U(t) is a semigroup generated by a strictly negative definite, self-adjoint unbounded operator A, such that A  − 1 is compact and f is of monotone type and is bounded by a polynomial and V (t) is a cadlag adapted stationary process.

Received 11/7/2011; Accepted 6/14/2012;Final 7/23/2012

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Authors and Affiliations

  1. Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

    Bijan Z. Zangeneh

Authors
  1. Bijan Z. Zangeneh
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Corresponding author

Correspondence to Bijan Z. Zangeneh .

Editor information

Editors and Affiliations

  1. , Department of Statistics, Purdue University, North University Street 150, West Lafayette, 47907, Indiana, USA

    Frederi Viens

  2. , Department of Mathematics, University of Kansas, Jayhawk Blvd 1460, Lawrence, 66045, Kansas, USA

    Jin Feng

  3. Dept. Mathematics, University of Kansas, Snow Hall 405, Lawrence, 66045-2142, Kansas, USA

    Yaozhong Hu

  4. , Department of Economics and Business, University Pompeu Fabra, Ramon Trias Fargas 25-27, Barcelona, 08005, Spain

    Eulalia Nualart 

Additional information

I would like to dedicate this paper to Professor David Nualart for his long lasting contribution to the field of stochastic analysis.

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Zangeneh, B.Z. (2013). Stationarity of the Solution for the Semilinear Stochastic Integral Equation on the Whole Real Line. In: Viens, F., Feng, J., Hu, Y., Nualart , E. (eds) Malliavin Calculus and Stochastic Analysis. Springer Proceedings in Mathematics & Statistics, vol 34. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5906-4_14

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