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PC analysis of stochastic differential equations driven by Wiener noise

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  • Le Maître, O.P.
  • Knio, O.M.

Abstract

A polynomial chaos (PC) analysis with stochastic expansion coefficients is proposed for stochastic differential equations driven by additive or multiplicative Wiener noise. It is shown that for this setting, a Galerkin formalism naturally leads to the definition of a hierarchy of stochastic differential equations governing the evolution of the PC modes. Under the mild assumption that the Wiener and uncertain parameters can be treated as independent random variables, it is also shown that the Galerkin formalism naturally separates parametric uncertainty and stochastic forcing dependences. This enables us to perform an orthogonal decomposition of the process variance, and consequently identify contributions arising from the uncertainty in parameters, the stochastic forcing, and a coupled term. Insight gained from this decomposition is illustrated in light of implementation to simplified linear and non-linear problems; the case of a stochastic bifurcation is also considered.

Suggested Citation

  • Le Maître, O.P. & Knio, O.M., 2015. "PC analysis of stochastic differential equations driven by Wiener noise," Reliability Engineering and System Safety, Elsevier, vol. 135(C), pages 107-124.
    • Handle: RePEc:eee:reensy:v:135:y:2015:i:c:p:107-124
    DOI: 10.1016/j.ress.2014.11.002
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    References listed on IDEAS

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    1. Crestaux, Thierry & Le Maıˆtre, Olivier & Martinez, Jean-Marc, 2009. "Polynomial chaos expansion for sensitivity analysis," Reliability Engineering and System Safety, Elsevier, vol. 94(7), pages 1161-1172.
    2. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
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    Cited by:

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    2. Pierre Étoré & Clémentine Prieur & Dang Khoi Pham & Long Li, 2020. "Global Sensitivity Analysis for Models Described by Stochastic Differential Equations," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 803-831, June.

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