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I have a continuous variable, $P_t$ whose evolution is unknown. However, I obtain a history of it i.e. $P_0, P_{dt}, P_{2dt}, ...... , P_T$. For a continuous process variable, I know that the rate of change can be written as $dlog(P_t)/d_t$ which is the analog to $(P_{t+1}-P_{t})/P_t$. So I write the mean of the rate of change from time 0 to time T as $$\frac{1}{T}\int_{0}^{T} y_{T-1}d\log(y_{T-1}).$$

However, what is the variance of the rate of change from time $0$ to time $T$. The analogue discrete version would be $var(P_t-P_{t-1})$ by treating $P_1-P_0, P_2-P_1......P_{t}-P{t-1}$ as a time series.

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