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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.

Kalman filter calculation for linear model.


e are considering equations of the form MATH MATH where $A,H,\rho,F,\phi$ are known time dependent deterministic matrixes, $x_{t}$ is observable at time $t$ random quantity, $\xi_{t}$ is a non observable random quantity that realized (determined) itself at $t$ , $w_{t-1}$ and $v_{t}$ are vectors of iid standard normal variables, realized at time $t$ and $g,h$ are known deterministic vectors. We will say that $X_{t}$ is the the observable part of the information and the $\QTR{cal}{F}_{t}$ is the total description at time $t$ . MATH

We start at time $t=0$ . We are given the distribution MATH and MATH . We assume that the distribution for the $\xi_{0}$ is normal: MATH where the $\eta_{1}$ is a vector of $\QTR{cal}{F}_{1}$ -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 MATH has the joint distribution MATH . Set $Y=AX+B$ for some square matrix $A$ and vector $B$ . Then $Y$ has the joint distribution MATH In particular, if MATH is a collection of iid standard normal variables with the joint distribution of $\xi$ is given by the function MATH then linear combination $\Sigma\xi+\mu$ with any non-degenerate matrix $\Sigma$ and vector $\mu$ has the joint distribution of MATH : MATH

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

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

We have MATH for $t=0$ . We calculate MATH with precision up to a multiplicative normalization constant: MATH We would like to put $F\left( y\right) $ to the form MATH for some symmetrical $Q$ . We are interested only in the knowledge of $a$ and $Q$ . Hence, MATH MATH MATH Therefore, MATH MATH The $Q$ is the inverse of the result's covariance matrix. We calculate it as follows MATH Using one of the forms for $Q^{-1}$ above we calculate $a$ as follows MATH We conclude MATH

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





Notation. Index. Contents.


















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