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.

## Orthogonal polynomials and quadrature rules.

efinition

(Quadrature rule) Let and are arbitrary numbers and is a measure on . We introduce the notations The "quadrature rule"

is said to have "degree of exactness" iff The "precise degree of exactness" is the maximal degree of exactness.

The quadrature rule with degree of exactness is called "interpolatory".

Definition

(Lagrange interpolation formula) For a set we define according to the relationship

 (Lagrange interpolation formula 1)

Note that hence Therefore Indeed, has the form . It is defined by values in distinct points.

Proposition

(Coefficients of quadrature rule) For a given set there exist such that the quadrature rule ( Quadrature rule 1 ) is interpolatory.

Proof

Integrate the formula ( Lagrange interpolation formula 1 ).

Proposition

(Higher degree of exactness) Given an integer , the formula ( Quadrature rule 1 ) has degree of exactness iff both of the following conditions are satisfied:

1. The formula ( Quadrature rule 1 ) is interpolatory.

2. The polynomial satisfies

Proof

(Necessity) The claim (1) follows immediately by the definition ( Quadrature rule ).

The claim (2) follows by substitution into the formula ( Quadrature rule 1 ): The term is zero because and term is zero because .

(Sufficiency) We assume (1),(2) and proceed to show that We divide by and represent the result as where and . Then The term is zero by (2). Thus and because and (1). We conclude since .

Proposition

(Gaussian quadrature rule) Let be the polynomials related to measure as in the definition ( Orthogonal polynomials ) and the inner product is positive definite. Then the formula ( Quadrature rule 1 ) has degree of exactness iff and are zeros of .

Proof

We apply the proposition ( Higher degree of exactness ). The condition (2) requires that and are zeros of by definition of . The numbers are all real by the proposition ( Zeros of orthogonal polynomials ).

The condition (1) becomes evident after noting similarity in structure of the polynomial and the polynomial of the definition ( Lagrange interpolation formula ).

The coefficients may be found from the proposition ( Coefficients of quadrature rule ).

 Notation. Index. Contents.