Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $T>0$
- $(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration of $\mathcal A$
- $\Phi^n,\Psi^n:\Omega\times[0,T]\to\mathbb R$ be $\mathcal F$-progressively measurable with $$\operatorname E\left[\int_0^t\Phi^n_s\:{\rm d}s\right],\:\operatorname E\left[\int_0^t\Psi^n_s\:{\rm d}s\right]<\infty\tag1$$ for $n\in\mathbb N$
- $(B^n)_{n\in\mathbb N}$ be an independent family of $\mathcal F$-Brownian motions on $(\Omega,\mathcal A,\operatorname P)$
Suppose $$X_t:=\sum_{n\in\mathbb N}\int_0^t\Phi_s^n\:{\rm d}B_s^n,\:Y_t:=\sum_{n\in\mathbb N}\int_0^t\Psi_s^n\:{\rm d}B_s^n\;\;\;\text{for }t\in[0,T]$$ exist in $L^2(\operatorname P,C^0([0,T])$.
How can we show that $$\operatorname E\left[X_tY_t\mid\mathcal F_r\right]=\operatorname E\left[\sum_{m,n\in\mathbb N}\left(\int_0^t\Phi_s^m\:{\rm d}B_s^m\int_0^t\Psi_s^n\:{\rm d}B_s^n\right)\mid\mathcal F_r\right]\tag2$$ for all $r,t\in[0,T]$ with $r\le t$?
In a more complicated setting, I've found $(2)$ here at the end/beginning of page 36/37.