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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.
V. Implementation tools.
1. Finite differences.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
A. Generation of random samples.
B. Acceleration of convergence.
C. Longstaff-Schwartz technique.
a. Normal Equations technique.
D. Calculation of sensitivities.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Normal Equations technique.

e wish to find MATH where the $\alpha$ and $\omega$ are integer indexes MATH , $\Omega\geq N$ and MATH is the indexed family of functions. This problem originates from the described in the previous section stochastic optimization procedure. We perform the standard minimization: MATH The last equation simplifies to the matrix problem MATH At this point we invoke the Singular Value Decomposition, see [Numerical] . There exist matrices $U,D,V$ such that MATH and $V$ is a square orthogonal matrix, $D$ is a diagonal matrix and $U$ is a matrix with orthogonal columns. Consequently, MATH Given the fact that the minimization problem is quadratic the last expression is the solution.

The above procedure breaks if the matrix A does not have a full rank.

Notation. Index. Contents.

Copyright 2007