# How to approximate expectation and variance of an integral from a discrete Time series financial dataset?

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?