Maximum likelihood estimation

Denote the $K$ observations of the process at times $t_k$ by $\Xi=(\xi_k)_{k=1\dots K}$. The probability of observation $\Xi$ given parameters $\Theta=(A,a,\Sigma)$ is

$$P(\Xi\vert\,\Theta) = \prod_{k=1}^K P(\xi_k,t_k\vert\,\xi_{k-1},t_{k-1})\,P(\xi_0)\quad ,$$ (74)

and, up to a parameter-independent constant, the log-likelihood function is

$$l(\Theta) = -{1\over 2}\sum_{k=2}^K\left[\det(V_{\Delta_k})... ...a_k}^- (\Delta\xi_k-\mu_{t_k\vert\,t_{k-1}}) \right]\quad ,$$ (75)

where $\Delta_k = t_k-t_{k-1}$ and $\Delta\xi_k = \xi_k-\xi_{k-1}$.