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Asymptotic behaviour of the density in a parabolic SPDE

Author

Listed:
  • Arturo Kohatsu
  • D. Márquez Carreras
  • M. Sanz Solé

Abstract

Consider the density of the solution $X(t,x)$ of a stochastic heat equation with small noise at a fixed $t\in [0,T]$, $x \in [0,1]$. In the paper we study the asymptotics of this density as the noise is vanishing. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable local integration by parts formula is developped.

Suggested Citation

  • Arturo Kohatsu & D. Márquez Carreras & M. Sanz Solé, 1999. "Asymptotic behaviour of the density in a parabolic SPDE," Economics Working Papers 371, Department of Economics and Business, Universitat Pompeu Fabra.
  • Handle: RePEc:upf:upfgen:371
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    References listed on IDEAS

    as
    1. Chenal, Fabien & Millet, Annie, 1997. "Uniform large deviations for parabolic SPDEs and applications," Stochastic Processes and their Applications, Elsevier, vol. 72(2), pages 161-186, December.
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    Keywords

    Malliavin Calculus; parabolic SPDE; large deviations; Taylor expansion of a density; exponential estimates of the tail probabilities; stochastic integration by parts formula;

    JEL classification:

    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General

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