I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prévôt/Röckner [PR07]:
$$ \int_0^T \langle \Psi \,\mathrm dW(t), \Phi(t)\rangle $$
A few useful properties are stated and a representation in terms of a "classical real valued Ito integral, namely $ \int_0^T \tilde{\Psi}_\Phi \, \mathrm dW(t)$, are given, under suitable assumption and for a suitably defined operator $\tilde{\Psi}_\Phi$ (see chapter 2 in [PR07]).
Does anyone have further references where I can find something more about it?
Moreover, is there any known formal results in the direction of proving that something like an integration by parts, as, for example
$$ \int_0^T \langle \mathrm dW(t), \Phi(t) \rangle = \int_0^T \langle W(t), -\mathrm d\Phi(t) \rangle + \langle W(t),\Phi(t) \rangle \Big|_0^T $$
can be performed?
Thanks
[PR07] A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, Vol. 1905 Claudia Prévôt and Michael Röckner, 2007.