Suppose $X_t$ satisfies the stochastic differential equation $dX_t = adt + dW_t$ ($a$ is a constant). What is the conditional distribution of $X_3$ given the values $X_0=0$, $X_1=-1$, $X_4=2$, and $X_5=1$?
My intuition says the process should be Gaussian but the only solution I've seen to such a problem is here, which uses a joint normal distribution approach. What are some other ways to approach a solution?