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I have discrete time series financial data, with time($u$), price($S$) and someVariable($q$) which looks something like this.

Time(u)       | Price(S) | someVariable(q)
12:00:00.021  |  21.75   |  0.987
12:00:00.373  |  21.85   |  0.952
12:00:00.579  |  21.60   |  0.92
12:00:00.930  |  21.15   |  0.84
12:00:01.126  |  20.25   |  0.81
12:00:01.841  |  20.85   |  0.79
...

Let's say there is some $X_{[0,T]}$ st.,

$X_{[0,T]} = \int_{0}^{T}(1-q_u)dS_u$, where $u \in [0,T]$

Now, I can compute $E[X_{[0.T]}]$ and $V[X_{[0.T]}]$ from this data with the above equation by Simpson's rule.

Alternatively, lets say I compute $E[q],E[dS],V[q],V[dS]$ from data in $u \in [0,T]$.

Now I'm very much interested in finding $E[X_{[0.T]}]$ and $V[X_{[0.T]}]$ as a function of $E[q],E[dS],V[q],V[dS]$.

Can the latter be approximated to give similar results as former?

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