Let we have a stochastic basis $(\Omega,\mathcal{F},\mathbb{F} = (\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ with standard assumptions. Let the integral $\int_0^t f(s,h) dB_s$, for $f(s,h)$ a progressively measurable process, $h \in \mathcal{H}$ - Banach space and $f(s,\cdot)$ is Frechet differentiable for all $s \geq 0$. Here $B = (B_t)_{t \geq 0}$ is an $\mathbb{F}$-Brownian motion.
Does anyone know about any result to switch the Frechet derivative (denoted as $D$ and applied to $h$ component) with the stochastic integral, as $$ D\int_0^t f(s,h) dB_s = \int_0^t Df(s,h) dB_s, $$ and what are the conditions? some like integrability and differentiability are obvious, boundedness is a fine candidate. I know about this result on the same problem with Lebesgue's integration, but stochastic one looks a bit more tricky.
P.S. I was trying to describe it as self-contained as possible without going into much details, sorry for the loopholes.