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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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Summary for Black equation in case q=1.


ummary

(Summary for Black 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 MATH according to the following procedure:

1. Calculate the diagonal matrix MATH

2. Calculate MATH

3. Do MATH Exit after step $\left( \&\right) $ in one of two following ways (a) or (c) or exit after step MATH in one of two following ways (b) or (d):

(a). If the condition number MATH is below 2 then set MATH

(b). If the Frobenius norm MATH is below certain precision then set MATH

(c). After 5 full iterations set MATH

(d). After 11 full iterations set MATH

The index sets $K_{n}$ come from considerations of the sections ( Rebalancing wavelet basis ) and ( Decomposition of payoff function in one dimension ). The calculation MATH may be done via GMRES.

There are situations when we need the above result in form of a single matrix that multiplies MATH to get MATH . For example, we need it when using variational inequalities or to use it as a partial preconditioner in a multidimensional situation. We provide a procedure for construction of such matrix next.

Summary

(Summary for Black equation in case q=1, inverted matrix) In context of the summary ( Summary for Black equation in case q=1 ) we transform from MATH to MATH via the multiplication MATH where the matrix MATH is calculated according to the following procedure.

1. Calculate the diagonal matrix MATH

2. Calculate MATH

3. Do MATH Exit after step $\left( \&\right) $ in one of two following ways (a) or (c) or exit after step MATH in one of two following ways (b) or (d):

(a). If the condition number MATH is below 2 then set MATH

(b). If the Frobenius norm MATH is below certain precision then $E$ is the answer.

(c). After 5 full iterations set MATH

(d). After 11 full iterations $E$ is the answer.

Due to its origins, we expect that the matrix $E$ has the property MATH The numerical experiment in the script blackGd.py confirms it. We use this property to remove stiffness due to finite time discretization as follows.

Summary

(Summary for Black equation in case q=1, any time interval) For a given $\Delta t_{n}$ and selected $d$ , we introduce MATH Let MATH be the matrix that comes from the summary ( Summary for Black equation in case q=1, inverted matrix ) with substitution MATH We set MATH and perform the multiplication MATH $d$ times, receiving MATH after the final multiplication. Such matrix should be used for the transformation MATH

Numerical experiment in the script blackGd.py shows that with $d=10$ we receive a precision of final result better than precision of adaptive wavelet approximation of the final payoff. Thus, the above summary completely removes all problems of finite time discretization. To be precise, we get numerical solution value MATH versus the value MATH that comes from analytical solution calculated with substitution MATH . We use a grid consisting of 62 functions approximating the final payoff with MATH -precision $1.0e-3$ .





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