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Assume that we have $\epsilon_1, \; \epsilon_2$ independent white noises.

  1. Can I write $\int_{0}^1 \epsilon_1^2(t)dt$

  2. Can I write $\int_{0}^1 \epsilon_1(t) \epsilon_2(t)dt$

1 and 2 obviously make no sense in $L^2$ nor in terms of Wiener integral. Is there any way I can make sense out of it?

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  • $\begingroup$ The answer provided by Carlo Beenakker was very helpful. However the articles provided do not cover the case of white noise multiplication - only powers. Does anyone have any clue? $\endgroup$
    – Sergiusz Wesolowski
    Commented Oct 18, 2016 at 4:10

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A consistent framework for "nonlinear stochastic calculus" has been developed in The square of white noise as a Jacobi field (2004), building on earlier work in Squared white noise and other non-Gaussian noises as Lévy processes on real Lie algebras (2002). A more pedestrian approach (perhaps more suited to your purpose) is taken in On powers of Gaussian white noise (2010), and Spectra for the product of Gaussian noises (2012).

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  • $\begingroup$ Is there a place to read an introductory and simple information White Noise? Specifically about its Auto Correlation Function? $\endgroup$
    – Royi
    Commented Jun 16, 2018 at 19:31

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