Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
1. Conditional probability.
2. Normal distribution.
A. Definition of normal variable.
B. Linear transformation of random variables.
C. Multivariate normal distribution. Choleski decomposition.
D. Calculus of normal variables.
E. Central limit theorem (CLT).
3. Brownian motion.
4. Poisson process.
5. Ito integral.
6. Ito calculus.
7. Change of measure.
8. Girsanov's theorem.
9. Forward Kolmogorov's equation.
10. Backward Kolmogorov's equation.
11. Optimal control, Bellman equation, Dynamic programming.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Multivariate normal distribution. Choleski decomposition.

uppose the random vector MATH is a collection of iid (independent identically distributed) standard normal variables. The joint distribution of $\xi$ is given by the function MATH The linear combination $\eta=\Sigma\xi+\mu$ with any non-degenerate matrix $\Sigma$ and vector $\mu$ is called "the multivariate normal variable MATH ". According to the results of the previous section, the joint distribution of MATH is given by the expression MATH

The matrix $A=\Sigma\Sigma^{T}$ is a positive-definite symmetric matrix and the covariance matrix of $\eta$ . In practice, one usually observes an approximation of $A$ from historical data and faces the problem of $\eta $ -reconstruction. There is more then one way to represent $A$ as a product of some matrix and its transpose. A particular representation MATH with a low-diagonal matrix $L$ MATH MATH is called "Choleski decomposition" of $A$ . The matrix $L$ may be reconstructed from $A$ through an efficient recursive procedure. See [Numerical] . An open source implementation of Cholesky decomposition is available in numpy.linalg module of Python programming language.

Notation. Index. Contents.

Copyright 2007