I'd like to estimate the *drift* of a continuous-paths, *non-stationary*, stochastic process $X\_t$ from a time series of values $\{X\_{i\Delta t}\}\_{i=1,\dots,N}$ sampled from a single realisation of that process over $t \in [0,T]$.

Although the objective is to calibrate a pricing model (specified under $\Bbb{Q}$) based on historical series (observed under $\Bbb{P})$, we can forget about this and assume we're working under a single measure $\Bbb{P}$.

I've bumped into a [similar question](https://quant.stackexchange.com/questions/2955/estimation-of-geometric-brownian-motion-drift), but unfortunately the references provided apply to processes admitting a *stationary distribution* (= linear drift Ornstein-Uhlenbeck processes used for interest rates modelling) an assumption which I'd like to move away from.

Actually, my assumptions could even be further simplified to the case of a simple Arithmetic Brownian motion with drift
$$ dX\_t = \mu dt + \sigma dW\_t $$

My question is: can someone point me towards a "nice" method to obtain an estimator $\hat{\mu}$ of $\mu$ from the observation of a $N$-sample $\{X\_i\}\_{i=1,...,N}$, where by "nice" I mean that the finite sample properties of the latter estimator should be better than the usual MLE (= LSE) estimator $$\hat{\mu} = \frac{1}{N-1} \sum\_{i=1}^{N-1} \Delta X\_i $$
whose relative error is proportional to $1/\sqrt{N\Delta t}$ i.e. to the time horizon $T=N \delta t$ over which the data sample is collected but not to the number of data points $N$ itself making it almost useless in practice.

I'm particularly interested in answers where the answerer has had a successful experience in implementing the method he/she proposes in practice. Because I've already come across different approaches/algorithms myself, but none were satisfying as far as I am concerned.