Let

- $T>0$
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$
- $B$ be a (standard, real-valued) $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$

Moreover, let $$\mathcal E^2_{\text{loc}}(H):=\left\{\Phi:\Omega\times[0,T]\to H:\Phi\text{ is }\mathcal F\text{-progressively measurable with }\operatorname P\left[\int_0^t\left\|\Phi_s\right\|_H^2{\rm d}s<\infty\right]=1\right\}$$ for any separable $\mathbb R$-Hilbert space $H$.

Let

- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be Borel measurable
- $\left\{\Phi^x:x\in\Lambda\right\}\subseteq\mathcal E^2_{\text{loc}}(\mathbb R)$ with $$\left(\Lambda\ni x\mapsto\Phi_t^x(\omega)\right)\in L^2(\Lambda)\;\;\;\text{for all }(\omega,t)\in\Omega\times[0,T]\tag 1$$
Moreover, let $$\Psi:\Omega\times[0,T]\to L^2(\Lambda)\;,\;\;\;(\omega,t)\mapsto\left(\Lambda\ni x\mapsto\Phi_t^x(\omega)\right)\;.$$ I want to show that $$\int_0^t\Phi_s^x\:{\rm d}B_s=\left(\int_0^t\Psi_s\:{\rm d}B_s\right)(x)\;\;\;\text{for all }x\in\Lambda\text{ and }t\in[0,T]\tag 2\;.$$

Is that possible, and if so, *how*? Of course, the usual approach would be to start with considering the case where each $\Phi^x$ is an elementary process (in the usual sense), but it's not trivial to show that then $\Psi$ is an elementary process (in the corresponding sense) too. So, maybe we need additional assumptions. Please take note of my related comment below.