# Average and Variance of the Rate of Change in a Continuous Variance

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.