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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.
8. Solving one dimensional mean reverting equation.
A. Reduction to system of linear algebraic equations for mean reverting equation.
B. Evaluating matrix R.
C. Localization for mean reverting equation.
D. Implementation for 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 mean reverting equation.


e use the summary ( Reduction to system of linear algebraic equations for q=1 summary ) with time-independent MATH : MATH

We calculate the matrix MATH : MATH MATH We evaluate the scalar product for all functions from MATH : MATH We perform integration by parts for the second term: MATH Hence, MATH We introduce notations MATH MATH MATH Then MATH

Summary

(Summary for mean reverting equation in case 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 and notation MATH is defined above. We use recipes of the section ( Summary for Black equation in case q=1 ) to invert the equation MATH . The index selections $K_{n}$ and time steps $\Delta t_{n}$ come from considerations of the sections ( Rebalancing wavelet basis ) and ( Decomposition of payoff function in one dimension ).





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