Covariance Quotes

This [covariance] is something that is not in the habit of thinking of most amateur investors. They look at their investments one at a time, and they don't, you always have to go back and say, what's the covariance? That's what really matters for what happen to your portfolio. Because when you invest in a lot of companies that are all the same, you're asking for trouble, because the whole thing is going to either blow up or succeed. And you can't live like that. You have to be looking for low covariance. (Robert J. Shiller)
A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly. View quote
Rudolf E. Kálmán