Conditional likelihood of continuously-combounded returns The simplest possible asset pricing model ist the geometric brownian motion for asset price. Here the price $S_t$ solve the familar
$$dS_t = (\mu +0.5 \sigma^2)S_t \, dt + \sigma S_t \, dW_t(\mathbb{P}).$$
where $\mu$ is the continuously-compounded expected return and $\sigma$ is the volatiliy. This model has a closed-form solution for continuously-compounded returns: $Y_t = \log(S_t/S_{t-1}) = \mu + \sigma \varepsilon_t$, with $\varepsilon_t \sim N(0,1)$.
The model generates a conditional likelihood for the vector of continuously-compounded returns of 
$$p(Y\mid\mu,\sigma^2) = \bigg( \frac{1}{\sqrt{2\pi\sigma^2}} \bigg)^T \exp \bigg \lbrace \frac{-1}{2\sigma^2} \sum_{t=1}^T(Y_t - \mu)^2 \bigg \rbrace,$$
and here is my question finally :), do i have to take for $Y = (S_1,\ldots,S_T)$ the equity prices of the underlying or the log returns $Y=(Y_1,\ldots,Y_T)$?
 A: What we have here is a stochastic process $\left(Y_t\right)_{t{}\geq{}0}$ consisting of independent, normally distributed random variables $Y_t{}\sim{}N\left(\mu\,,\,\sigma^2\right)$. As you indicated, these are the "rates" or logs of continuously compounded returns over non-overlapping (hence the independence) unit intervals of time. Consequently, the likelihood function of a finite sample of these, say realisations $y_1,\ldots,y_T$, gives the density associated with observing this sequence of log returns as a function of the parameters $\mu$ and $\sigma$: it is a product of the probability density functions of the respective log returns. That is, as you stated, the likelihood function is:
$$
p(y_1,\ldots,y_T\mid\mu,\sigma^2){}={}\bigg( \frac{1}{\sqrt{2\pi\sigma^2}} \bigg)^T \exp \bigg \lbrace \frac{-1}{2\sigma^2} \sum_{t=1}^T(y_t - \mu)^2 \bigg \rbrace\,,
$$ 
where I have simply emphasized the fact that it is the realisations of the log returns that are important (of course, these are computed from the observed values of the underlying equity).
