The discrete Kalman Filter (assuming a linear time invariant system with uncorrelated Gaussian process and sensor noise and finite observation interval) solves the problem for minimizing the mean square error at every time step (the trace of the a posteriori covariance matrix). $$\text{min}E\left[(x_k-\hat{x_k})^T(x_k-\hat{x_k}) \right] $$ While reading A Study of the Duality between Kalman Filters and LQR Problems section 4 it is shown in proposition 5 that the filter also solves $$\text{min } \text{Tr}(P_N W_f) + \sum_k^{N-1} \text{Tr}\left(P_k W \right) $$ where W is some weighing matrix and $P$ is covariance.
In the continuous Kalman filter case it is my understanding that the filter also minimizes the trace of $P$ for every time t (or the mean square error). In Dan Simon's book Optimal State estimation it states that the continuous Kalman filter minimizes $$J_e = \int_0^{t_f} E[(x-\hat{x})^T(x-\hat{x})]dt$$
I was wondering if anyone can point/explain me to some additional reading for the discrete and especially continuous case explaining how the sum/integral measures are derived or are equivalent to minimizing the mean square error.
Thanks