Consider a Brownian Motion with drift, $X$, on the interval $[0; T]$ given by $$dX_t = \mu dt + \sigma dW_t.$$
Suppose that the interval is split into $n$ pieces of equal size to define $\Delta:=T/n$ at time points $t_i, \; i=0,...,n$. Assume we have observed $X$ at these points. That is, we have a sequence of observed values $(x_i)_{i=0,...,n}$.
I want to estimate $\mu$ and $\sigma$ from the data. How can I do this?
My first approach is to look at the distribution of $X_t$. It is $$X_t \sim N(\mu t, \sigma^2 t)$$ as I use that $W_t \sim N(0, t)$. However, this does not seem to help me estimate $\mu$ and $\sigma$ - as I would use Maximum Likelihood Estimation (MLE) to find the mean and variance of $X_t$ (but these are not even identically distributed... they are time-dependtent).