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.

## Convolution of normal distributions.

efore we proceed with derivations for a linear version of the Kalman filter we perform some general calculation we will use repeatedly. We would like to evaluate the following integral where the are matrixes and are vectors. We calculate The -integral is evaluated via completion of the square: for some . Hence, we require The expression is positive and symmetrical, therefore such exists. We continue We have We perform the change in the integral. The Jacobian is . The integral is We collect our results together, Observe that if is such that then with Hence, Therefore is the mean of the resulting normal distribution. Consequently, we introduce such that and calculate The is a quadratic function and we already know that , hence We have

Hence, the covariance of the result is the inverse of the above expression: Therefore, we may state the result

 Notation. Index. Contents.