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 where the and are integer indexes , and is the indexed family of functions. This problem originates from the described in the previous section stochastic optimization procedure. We perform the standard minimization: The last equation simplifies to the matrix problem At this point we invoke the Singular Value Decomposition, see [Numerical] . There exist matrices such that and is a square orthogonal matrix, is a diagonal matrix and is a matrix with orthogonal columns. Consequently, 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.