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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.
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 $t$ is discrete: $t=1,2,...$ . Suppose the random variables $x_{t}\in R^{M}$ and $\xi_{t}\in R^{N}$ are given by the recursive relations MATH MATH where the MATH are some deterministic functions of time $t$ of appropriate dimensionality. As usual the $\QTR{cal}{F}_{t}$ denotes the algebra of events representing the amount of information, available at time $t.$ The Gaussian random variables $w_{t}$ , $\nu_{t}$ are MATH measurable and independent from all $\QTR{cal}{F}_{t}-$ measurable variables of this setup. We understand such situation as variables $w_{t}$ , $\nu_{t}$ being in the future of the time moment $t$ . We will use this property heavily without further reference. The variables $w_{t}$ , $\nu_{t}$ have zero mean and known covariance matrixes MATH The variable $x_{t}$ is observable. As time progresses we accumulate values MATH . The variable $\xi_{t}$ is not observable.

We introduce notation $\hat{\xi}_{t,s}$ for estimation of value $\xi_{t}$ based on information $\QTR{cal}{F}_{s}$ . Denote $\hat{x}_{t,s}$ forecast of $x_{t}$ based on $\QTR{cal}{F}_{s}.$ Also, MATH .

Initially, at time $t=1$ , we are given values MATH and $P_{1,1}$ . Our goal is to determine recursively the quantities $\hat{\xi }_{t,t}$ , $P_{t,t}$ as the information MATH 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.


















Copyright 2007