If we have an AR(1) process, i.e: $X_{t+1} = \alpha X_t + e_{t+1}$ with $X_0=0$ then what is its Markov Chain transition density? We know that for a Markov chain, the following holds:
$P(X_{t+1}\leq x_{t+1} \mid X_0\leq x_0, X_1 \leq x_1, ... ) = P(X_{t+1}\leq x_{t+1} \mid X_t\leq x_t)$
For an AR(1) process this should then be
$P(\alpha X_{t} + e_{t+1}\leq x_{t+1} \mid X_t\leq x_t)$
From what I understand, the transition density $q(x_{t+1}\mid x_t)$ is the derivative of $P(\alpha X_{t} + e_{t+1}\leq x_{t+1} \mid X_t\leq x_t)$. But how should I take the derivative...?