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 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 is a collection of iid (independent identically distributed) standard normal variables. The joint distribution of is given by the function The linear combination with any non-degenerate matrix and vector is called "the multivariate normal variable ". According to the results of the previous section, the joint distribution of is given by the expression

The matrix is a positive-definite symmetric matrix and the covariance matrix of . In practice, one usually observes an approximation of from historical data and faces the problem of -reconstruction. There is more then one way to represent as a product of some matrix and its transpose. A particular representation with a low-diagonal matrix is called "Choleski decomposition" of . The matrix may be reconstructed from 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.