Kalman filter by hand

I am now learning the Kalman filter and wants to implement it by hand to understand it better. To be specific I want to first simulate a sequence of data by $$\dot x=Ax+Bw\\ y=Cx+Dv,\\ E[x(0)]=0,E[x(0)x'(0)]=P_0,$$ where $w,v$ are white noise processes with unit variance. Then I need to construct the Kalman filter by $$\dot{\hat x}=A\hat x+QC'(y-C\hat x), \hat x(0)=0$$ where $Q$ solves the ricatti differential equation $$\dot Q=AQ+QA'-QC'CQ+C'C,Q(0)=P_0.$$ I have two questions:

1. How do I generate the sequence of data governed by a differential equation?

2. How do I solve the ricatti differential equation?

I know this post shows no effort, but I am new to mathematica and have no idea where to start (especially that I cannot generate the data needed, which stops me from any progress). Any hint is appreciated.

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If you have version 8 or higher, look up RicattiSolve‌​, KalmanEstimator and the general guide to control systems in the documentation. – Verbeia Apr 7 '13 at 0:55
Regarding 1. you could use NestList. – b.gatessucks Apr 7 '13 at 7:01
A pre-v9 related Wolfram Library Archive entry: Kalman Filter. – István Zachar Apr 7 '13 at 7:41