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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.
V. Stochastic optimization in one dimension.
1. Review of variational inequalities in maximization case.
2. Penalized problem for mean reverting equation.
3. Impossibility of backward induction.
4. Stochastic optimization over wavelet basis.
A. Choosing probing functions.
B. Time discretization of penalty term.
C. Implicit formulation of penalty term.
D. Smooth version of penalty term.
E. Solving equation with implicit penalty term.
F. Removing stiffness from penalized equation.
G. Mix of backward induction and penalty term approaches I.
H. Mix of backward induction and penalty term approaches I. Implementation and results.
I. Mix of backward induction and penalty term approaches II.
J. Mix of backward induction and penalty term approaches II. Implementation and results.
K. Review. How does it extend to multiple dimensions?
VI. Scalar product in N-dimensions.
VII. Wavelet transform of payoff function in N-dimensions.
VIII. Solving N-dimensional PDEs.
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Mix of backward induction and penalty term approaches II.


e continue research of the previous section ( Mix of backward induction and penalty term approaches I. Implementation and results ).

We start from a function $z$ and aim to construct MATH

For an initial $d_{0}$ , form the set MATH s.t. MATH

Calculate MATH

Find MATH

Set MATH where the normalization parameter $\theta$ is derived from the requirements MATH Thus MATH MATH MATH

We calculate the components. Let MATH then

MATH We apply the operation MATH to $\left( \&\right) $ and obtain MATH Let MATH then MATH where MATH is $k^{\ast}$ -th row of the matrix $\bar{G}_{d^{\ast}}$ , transposed into a column.

Summary

(Constructing maximum) We start from columns $a$ and $c$ s.t. MATH and construct a column $c^{\ast}$ s.t. MATH via the following procedure.

Choose an initial scale $d_{0}$ s.t. the distance between any two singular roots of $g-z$ is greater than $2^{-d_{0}}$ . Form the set MATH s.t. MATH where MATH

1. Calculate MATH where MATH

2. Find MATH and exclude from $\Phi$ all $\left( d,k\right) $ s.t. $\kappa_{d,k}\leq0$ and exclude MATH . If $\Phi$ is empty then set MATH and MATH

3. Calculate MATH where MATH

4. Set MATH where MATH and MATH is $k^{\ast}$ -th row of the matrix $\bar{G}_{d^{\ast}}$ , transposed into a column.

5. Exit or go to 1.

The procedure is adapted to parallel architecture because one can subtract several functions with non-overlapping support. Most intensive pieces of calculation may be pre-calculated.

An adaptive extension of the procedure would involve selecting MATH and $\eta_{d_{1},k_{1}}$ from two different classes. Indeed, MATH should be adapted to subtract biggest piece from the solution. The functions $\eta_{d_{1},k_{1}}$ should be designed not to allow a change of sign.





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