Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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 MATH be a non-decreasing function such that the limits MATH exist and are finite and the moments MATH exist and are finite for all $r=0,1,2,...$ and $\mu_{0}>0.$

2. Let $\QTR{cal}{P}_{d}$ be the space of all polynomials of degree not greater then $d$ . Let MATH Let $\QTR{cal}{P}$ be the space of all polynomials.

3. We introduce the notation MATH

4. The inner product MATH is said to be "positive definite" if MATH is positive for all $u\in\QTR{cal}{P}$ , $u\not \equiv 0.$

In general we have MATH . We might have MATH for some $u\not \equiv 0$ if, for example, MATH , $w\geq0$ and MATH for MATH .

Proposition

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

MATH (Hankel determinants)

Proof

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

Since $M_{n}$ is a symmetric matrix, we have a decomposition MATH where $Q$ is an orthogonal matrix and $\Lambda_{n}$ is a diagonal matrix. Hence, $M_{n}$ is positive definite iff $\Lambda_{n}$ is positive definite. Since $\det Q=1$ (orthogonal matrix) we have MATH and MATH Thus $\Lambda_{n}$ is positive definite for all $n$ iff MATH are positive for all $n.$

Definition

(Orthogonal polynomials)

1. We introduce the notation MATH and MATH for polynomials with the following properties: MATH for all $k,p$ and some MATH .

2. We introduce the notation MATH and MATH for the polynomials MATH

Proposition

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

Proof

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

Proposition

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

1. MATH is a basis in $\QTR{cal}{P}_{d},$

2. MATH is a basis in $\QTR{cal}{P}$ .

Proof

$\QTR{cal}{P}_{d}$ is $\left( d+1\right) $ -dimensional and MATH are linearly independent, hence (1). MATH hence (2).





Notation. Index. Contents.


















Copyright 2007