Consider the time-evolution operator $U(dt) = \exp(-id tJH_0 - i d B_t V)$ where $H_0$ and $V$ are some Hamiltonians, and $J$ is some coupling. $B_t$ is the Brownian motion on $\mathbb{R}$ with $B_0 = 0$. From physically motivated conditions, the coupling must be rescaled as $J = \sqrt{\gamma /dt}$, and after replacing this in $U(dt)$, I end up with terms of the form
$$ U(d t) = I - i\sqrt{\gamma dt}H_0 - id B_t V - \frac{1}{2}[\gamma dt H_0^2 + d t V^2 + dB_t \sqrt{dt} (VH_0 + H_0V)],$$
where I have used the usual rules $d t^2 = 0, d B_t^2 = dt,$ etc. Can I take $dB_t\sqrt{d t} = 0$? I have searched for products of that form, but I have not found any.