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I. Wavelet calculations.
II. Calculation of approximation spaces in one dimension.
III. Calculation of approximation spaces in one dimension II.
IV. One dimensional problems.
1. Decomposition of payoff function in one dimension. Adaptive multiscaled approximation.
2. Constructing wavelet basis with Dirichlet boundary conditions.
3. Accelerated calculation of Gram matrix.
4. Adapting wavelet basis to arbitrary interval.
5. Solving one dimensional elliptic PDEs.
6. Discontinuous Galerkin technique II.
7. Solving one dimensional Black PDE.
A. Example Black equation parameters.
B. Reduction to system of linear algebraic equations for Black PDE.
C. Adaptive time step for Black PDE.
D. Localization.
E. Reduction to system of linear algebraic equations for q=1.
F. Preconditioner for Black equation in case q=1.
G. Summary for Black equation in case q=1.
H. Implementation of Black equation solution.
8. Solving one dimensional mean reverting equation.
V. Stochastic optimization in one dimension.
VI. Scalar product in N-dimensions.
VII. Wavelet transform of payoff function in N-dimensions.
VIII. Solving N-dimensional PDEs.
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Reduction to system of linear algebraic equations for q=1.


e spell out results of the section ( Reduction to system of linear algebraic equations for Black PDE ) in case $q=1$ (piecewise constant in time). The system of linear equations takes the form MATH MATH MATH MATH MATH MATH We multiply the equation MATH with MATH and obtain MATH

Summary

(Reduction to system of linear algebraic equations for q=1) We start from a column MATH given by MATH For $n=N-1,N-2,...,0$ solve MATH where MATH MATH MATH MATH MATH

The index sets $K_{n}$ come from considerations of the sections ( Rebalancing wavelet basis ) and ( Decomposition of payoff function in one dimension ).

The above matrix MATH is poorly conditioned. The summary is not a practical recipe. The following section contains practical derivations.





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