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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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Localization.


he interval $\left[ A,B\right] $ was inserted into the problem ( Example Black problem ) to avoid considering a PDE problem on the entire real line. We constructed a basis of functions with finite support, unsuitable for finite element calculation on unbounded set. We would like to place the numbers $A,B$ to positions according to the rule MATH for some number $x_{0}>0$ . The process $X_{t}$ is given by the SDE MATH We calculate MATH MATH MATH The distribution density $p_{\xi}$ of $\xi$ is a symmetric function MATH Thus, we take $\xi$ as a controlling parameter: MATH for a fixed positive number $\xi_{0}$ .

The only remaining step is to make sure that the interval $\left[ A,B\right] $ is adapted to binary structure of wavelet basis. We choose a scale parameter $d_{0}$ then MATH

Summary

(Choice of interval) The choice of interval $\left[ A,B\right] $ is controlled by two parameters $\xi_{0}$ and $d_{0}$ . The $\xi_{0}$ controls width of the interval expressed in numbers of sigma. The $d_{0}$ controls scale.

We calculate MATH MATH where $x_{0}$ is the time- $t$ position of the process $X_{t}$ .

The parameter $\xi_{0}$ should be taken so that MATH would be comparable to precision of initial approximation of the payoff function, discussed in the section (Decomposition of payoff function in one dimension ). The parameter $d_{0}$ should be taken to be the minimal number needed to provide the condition ( Sufficiently fine scale 2 ).





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