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.

## Smooth version of penalty term.

n the previous section we argued that we must have implicit form of the penalty term:

 (Implicit equation with penalty term)
We arrived to a non-linear equation. The non-linear part is where the scalar products take both positive and negative values. We do not have convexity. Then, to have an efficient way of finding a solution, we must have at least a continuous matrix of first derivatives. Hence, we pick up the idea of substitution already introduced in the section ( Time discretization of penalty term ). The function must have the following properties: It suffices to set for a parameter . Taking substantially above 1 introduces uneven weights to values in the area and, thus, contributes to stiffness. However, taking significantly above 1 also increases smoothness and provides access to high order calculational techniques.