This may be a strange question, but I was curious about it. Under the discounted measure, one can write the lognormal price dynamics as:
$\frac{dS}{S} = \sigma dW$.
It is also Markov. Why is it that despite being Markov, it is somewhat industry-standard practice to use historical data to measure $\sigma$?
First of all , the discounted stock price is a martingale, not the stock itself. $$dD(t)S(t)=\sigma D(t)S(t)dW(t)$$, under the risk neutral measure(what you called "discounted measure"), and $D$ the discounting process.
This measure is tied to a world where you are risk-averse . In other words, I can have a position, and remove the risk by taking another position. Therefore, the market tells you where $\sigma$ is, not its past or future. This world guarantees non-arbitrage prices. It makes no sense to use this measure and value $\sigma$ by using an historical estimate, a simple look at any time series will tell you that $S_t$ is not a martingale...