I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 1 Time Series.
 A. Time series forecasting.
 B. Updating a linear forecast.
 C. Kalman filter I.
 a. Kalman filter computation at t=1.
 b. Kalman filter computation for general t.
 c. Calibration of parameters with Kalman filter.
 D. Kalman filter II.
 E. Simultaneous equations.
 2 Classical statistics.
 3 Bayesian statistics.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Kalman filter I.

his section follows [Hamilton] . The time is discrete: . Suppose the random variables and are given by the recursive relations where the are some deterministic functions of time of appropriate dimensionality. As usual the denotes the algebra of events representing the amount of information, available at time The Gaussian random variables , are measurable and independent from all measurable variables of this setup. We understand such situation as variables , being in the future of the time moment . We will use this property heavily without further reference. The variables , have zero mean and known covariance matrixes The variable is observable. As time progresses we accumulate values . The variable is not observable.

We introduce notation for estimation of value based on information . Denote forecast of based on Also, .

Initially, at time , we are given values and . Our goal is to determine recursively the quantities , as the information flows in.

 a. Kalman filter computation at t=1.
 b. Kalman filter computation for general t.
 c. Calibration of parameters with Kalman filter.
 Notation. Index. Contents.