I am looking for a basic statement of stochastic Fubini's theorem for Brownian Motion and simple integrands. I have been searching in the internet but I have only been able to find references which are overly academic in nature.
The setup is a probability space $(\Omega,\mathscr{F},\mathbb{P})$ where $\mathscr{F}=(\mathscr{F}_t:t \in \mathbb{R}^+)=\sigma(W_t: t \in \mathbb{R}^+)$ is the filtration generated by a Brownian Motion $(W_t: t \in \mathbb{R}^+)$.
We now consider:
- A generic process $F(u,v) \equiv F(u,v,\omega)$, and
- A deterministic function $f(u,v)$,
where $(u,v) \in \mathbb{R}^2$ and $\omega \in \Omega$.
Q.1) Under what conditions can we invert the order of integration for a stochastic integrand: $$\int_0^{t_1}\left(\int_0^{t_2}F(u,v)\text{d}W_u\right)\text{d}v=\int_0^{t_2}\left(\int_0^{t_1}F(u,v)\text{d}v\right)\text{d}W_u \ \ ?$$
Q.2) Under what conditions can we invert the order of integration for a deterministic integrand: $$\int_0^{t_1}\left(\int_0^{t_2}f(u,v)\text{d}W_u\right)\text{d}v=\int_0^{t_2}\left(\int_0^{t_1}f(u,v)\text{d}v\right)\text{d}W_u \ \ ?$$
Q.3) Is there any difference w.r.t. to the answer to Q.1 and Q.2 above if $F$ or $f$ only depend on one variable, either $u$ or $v$?
The conditions might be on $F(u,v)$, $f(u,v)$, $t_1$, $t_2$ (e.g. adaptedness w.r.t to $\mathscr{F}$, boundness, etc.).
Any non-academic oriented reference is greatly appreciated; any explicit answer to the questions is also greatly appreciated.