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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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 MATH where the $A,\sigma,\omega$ are matrixes and $a,b,x,y$ are vectors. We calculate MATH The $y$ -integral is evaluated via completion of the square: MATH for some MATH . Hence, we require MATH The expression MATH is positive and symmetrical, therefore such $\alpha$ exists. We continue MATH We have MATH We perform the change $z=\alpha y-\beta$ in the integral. The Jacobian is $dy/dz=\alpha^{-1}$ . The integral is MATH We collect our results together, MATH MATH Observe that if $x^{\ast}$ is such that MATH then MATH with MATH Hence, MATH Therefore MATH is the mean of the resulting normal distribution. Consequently, we introduce $y$ such that MATH and calculate MATH MATH MATH MATH The $F\left( y\right) $ is a quadratic function and we already know that MATH , hence MATH We have MATH MATH MATH MATH

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

Notation. Index. Contents.

Copyright 2007