Let us consider a random walk:
$$x_{t+1} = x_t + \sigma e_{t+1}, x_0 = 0.$$
Let us consider the observable random variable:
$$y_t = x_t + \nu \eta_t.$$
We assume that $(e_t)$ and $(\eta_t)$ are independent, iid, and standard Normal.
Can we explicitly find the maximum likelihood estimators of $\sigma$ and $\nu$ (we only observe the $y$s)?