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Quantitative Analysis
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
1. Calculational Linear Algebra.
2. Wavelet Analysis.
A. Elementary definitions of wavelet analysis.
B. Haar functions.
C. Multiresolution analysis.
D. Orthonormal wavelet bases.
E. Discrete wavelet transform.
a. Recursive relationships for wavelet transform.
b. Properties of sequences h and g.
c. Approximation and detail operators.
F. Construction of MRA from scaling filter or auxiliary function.
G. Consequences and conditions for vanishing moments of wavelets.
H. Existence of smooth compactly supported wavelets. Daubechies polynomials.
I. Semi-orthogonal wavelet bases.
J. Construction of (G)MRA and wavelets on an interval.
3. Finite element method.
4. Construction of approximation spaces.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Properties of sequences h and g.


roposition

(Integral of scaling function) Let MATH is a scaling function of an MRA. Then MATH

Proof

By the definition ( Multiresolution analysis )-1,2 we have MATH We calculate the last term using the proposition ( Basic properties of Fourier transform )-3: MATH We use the formula ( Property of scale and transport 6 ): MATH Using the propositions ( Dominated convergence theorem ) and ( Fubini theorem ), MATH The collection MATH is an orthonormal basis in MATH . We use the proposition ( Parseval equality ). MATH We use the proposition ( Basic properties of Fourier transform )-4,5. MATH Thus MATH

Proposition

(Integral of wavelet) Let MATH is a scaling function of an MRA and $\psi$ is the associated wavelet, see the proposition ( Scaling equation 2 ). We have MATH

Proof

According to the proposition ( Scaling equation 2 ), MATH Thus MATH According to the proposition ( Scaling equation 3 ), MATH Thus MATH where MATH According to the proposition ( Scaling equation ), MATH an according to the proposition ( Integral of scaling function ) MATH thus MATH

Proposition

(Properties of sequences h and g) We assume the setup ( Discrete wavelet transform ), MATH and $\phi$ is real-valued. Then

(a) MATH ,

(b) $\sum_{k}g_{k}=0$ ,

(c) MATH ,

(d) MATH ,

(e) MATH ,

(f) MATH .

Proof

of (a). Within the proof the proposition ( Integral of wavelet ) we saw MATH

Proof

of (b). According the proposition ( Integral of wavelet ), MATH Change of variable, $2x-k\Rightarrow2x$ and then $2x\Rightarrow x$ . MATH We use the proposition ( Integral of scaling function ). MATH

Proof

of (c),(d),(e). We use orthonormality of MATH for every $s$ : MATH We use the formula ( Property of scale and transport 7 ). MATH The property (d) follows similarly from orthonormality of MATH for every $s$ , see the proposition ( Existence of orthonormal wavelet bases 2 ). The property (e) follows by considering MATH , see the proposition ( Existence of orthonormal wavelet bases 1 )-c.

Proof

of (f). We combine the proposition ( Recursive relationships for wavelet transform )-c with ( Recursive relationships for wavelet transform )-a,b: MATH We now express $c_{1,n}$ and $d_{1,n}$ in terms of $c_{0,k}$ using ( Recursive relationships for wavelet transform )-a,b. MATH Thus MATH





Notation. Index. Contents.


















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