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?



can I integrate product or square of a white noise in any sense?
A consistent framework for "nonlinear stochastic calculus" has been developed in [The square of white noise as a Jacobi field](https://arxiv.org/abs/math/0401370v1) (2004), building on earlier work in [Squared white noise and other non-Gaussian noises as Lévy processes on real Lie algebras](https://art.torvergata.it/retrieve/handle/2108/75107/147446/AcFrSk00.pdf) (2002). A more pedestrian approach (perhaps more suited to your purpose) is taken in [On powers of Gaussian white noise](http://arxiv.org/abs/1010.2992) (2010), and [Spectra for the product of Gaussian noises](http://arxiv.org/abs/1207.1881) (2012).


