Discrete Stochastic Processes and Optimal Filtering

Appendix A

Resolution of Riccati's equation

Let us show that:

Let us take again the developed expression of the covariance matrix of the prediction error of section 7.3.6

with:

and:

By replacing K(K) and Cov I_K, by their expressions, in the recursive writing of, , we are going to be able to simplify the expression of the covariance matrix of the prediction error.

To lighten the expression, we are going to eliminate the index K when there is no ambiguity by noting , and I = I_K

Thus:

i.e. in replacing K with its expression.

The 3^rd and 6^th term cancel each other out and the 4^th and 7^th term ...