So if I have the stochastic differential equation
$dX(t) = \mu dt + \sigma dW(t)$,
I know the maximum likelihood estimate for the drift and volatility is:
\begin{align} &\hat{\hat{\mu} = \frac{1}{N\Delta t}\sum_{i=0}^{N-1}} X(t_{n+1}) - X(t_n)\\ &\hat{\sigma}^2 = \frac{1}{\Delta t (N-1)}\sum_{i=0}^{N-1} (X(t_{n+1}) - X(t_n) - \hat{\mu}\Delta t)^2 \end{align} However my question is if the equation changes slightly
$dX(t) = \mu X(t) dt + \sigma X(t) dW(t)$,
are the previous estimates of drift and volatility consistent with this new equation?
I found my question here : Parameter estimation for Stochastic differential equation , and it seems yes, but it is not explicitly said.
In general if I have a process that satisfies:
$dX(t) = f_1(X(t),\mu)dt + f_2(X(t),\sigma) d W(t)$, are the above estimates always valid for approximating the drift and volatility parameters (assuming that $f_1,f_2$ smooth functions).
Thank you