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.
 D. Kalman filter II.
 a. General Kalman filter problem.
 b. General Kalman filter solution.
 c. Convolution of normal distributions.
 d. Kalman filter calculation for linear model.
 e. Kalman filter in non-linear situation.
 f. Unscented transformation.
 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 calculation for linear model.

e are considering equations of the form where are known time dependent deterministic matrixes, is observable at time random quantity, is a non observable random quantity that realized (determined) itself at , and are vectors of iid standard normal variables, realized at time and are known deterministic vectors. We will say that is the the observable part of the information and the is the total description at time .

We start at time . We are given the distribution and . We assume that the distribution for the is normal: where the is a vector of -measurable iid standard normal variables.

We will be repeatedly using the following result (see ( Linear transformation of random variables )).

Proposition

Suppose the vector of random variables has the joint distribution . Set for some square matrix and vector . Then has the joint distribution In particular, if is a collection of iid standard normal variables with the joint distribution of is given by the function then linear combination with any non-degenerate matrix and vector has the joint distribution of :

The summary of the procedure is as follows. We have the distribution from previous steps and from the model. We calculate through steps

The is the normalization term. We do not need to calculate it explicitly. The distributions come from the model. The integral is calculated using the result of the previous section:

We have for . We calculate with precision up to a multiplicative normalization constant: We would like to put to the form for some symmetrical . We are interested only in the knowledge of and . Hence, Therefore, The is the inverse of the result's covariance matrix. We calculate it as follows Using one of the forms for above we calculate as follows We conclude

Following the outlined procedure we now calculate where the we have just obtained and the comes from the model Hence, we apply the formula for the convolution of normal distributions We have The recursion is completed.

 Notation. Index. Contents.