I have a time series of values from a Gaussian white noise which are evenly spaced ($\Delta t$). I'd like to approximate $$\int_{0}^{T}\mathsf{A}(t)\mathbf{f}(t) \,\mathrm{d}t$$ where $\mathbf{f}$ is the Gaussian white noise, "A" arbitrary. In my case, I have a discrete set of N realisations of this, and so my first thought was to numerically integrate with $$\sum_{i=0}^{N}\mathsf{A}(t_i)\mathbf{f}(t_i)\Delta t$$ I've seen some examples similar to this online, but the world of SDEs is hinting that I should, in fact, be writing $$ \sum_{i=0}^{N}\mathsf{A}(t_i)\mathbf{f}(t_i)\sqrt{\Delta t}$$ based on the idea that white noise integrates to give Brownian/red noise, and the properties of Wiener processes should then be followed.
I'm no expert in this area, so which (if either) is the correct choice, and, intuitively, why?
Bonus question!
My data are of course not truly "white" at all time scales, and have a decorrelation time, say $\lambda$. How can I correctly include this so that any time units drop out of the result? If I should use $\sqrt{\Delta t}$, should it actually be $\sqrt{\frac{\Delta t}{\lambda}}$?