I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 2 Gauss-Hermite Integration.
 A. Gram-Schmidt orthogonalization.
 B. Definition and existence of orthogonal polynomials.
 C. Three-term recurrence relation for orthogonal polynomials.
 D. Orthogonal polynomials and quadrature rules.
 E. Extremal properties of orthogonal polynomials.
 F. Chebyshev polynomials.
 3 Asymptotic expansions.
 4 Monte-Carlo.
 5 Convex Analysis.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Definition and existence of orthogonal polynomials.

efinition

(Positive definite inner product)

1. Let be a non-decreasing function such that the limits exist and are finite and the moments exist and are finite for all and

2. Let be the space of all polynomials of degree not greater then . Let Let be the space of all polynomials.

3. We introduce the notation

4. The inner product is said to be "positive definite" if is positive for all ,

In general we have . We might have for some if, for example, , and for .

Proposition

(Criteria of positive definiteness) The inner product is positive definite iff where

 (Hankel determinants)

Proof

Let so that then Thus is positive definite iff are positive definite.

Since is a symmetric matrix, we have a decomposition where is an orthogonal matrix and is a diagonal matrix. Hence, is positive definite iff is positive definite. Since (orthogonal matrix) we have and Thus is positive definite for all iff are positive for all

Definition

(Orthogonal polynomials)

1. We introduce the notation and for polynomials with the following properties: for all and some .

2. We introduce the notation and for the polynomials

Proposition

(Existence of orthogonal polynomials) If the inner product is positive definite then there exists a sequence .

Proof

Apply Gram-Schmidt orthogonalization (see the section ( Gram-Schmidt orthogonalization )) to .

Proposition

(Basic property of orthogonal polynomials) If inner product is positive definite then

1. is a basis in

2. is a basis in .

Proof

is -dimensional and are linearly independent, hence (1). hence (2).

 Notation. Index. Contents.