Mathematical framework for pseudo-spectra of linear stochastic difference equations

• Autores: Andrés Bujosa Brun, Marcos Bujosa Brun, Antonio García Ferrer
• Localización: Documentos de Trabajo (ICAE), ISSN-e 2341-2356, Nº. 13, 2013, págs. 1-13
• Idioma: inglés
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• Resumen

  • Although spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not always so for the non-stationary case. Here, we establish a sound mathematical framework for the spectral analysis of non-stationary solutions of linear stochastic difference equations. To achieve it, the classical problem is embedded in a wider framework, the Rigged Hilbert space; the Fourier Transform is extended, and a new Extended Fourier Transform pair pseudocovariance function/pseudo-spectrum is defined. Our approach is an extension proper of the classical spectral analysis, where the Fourier Transform pair auto-covariance function/spectrum is a particular case, and consequently spectrum and pseudo-spectrum are identical when the first one is defined.