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.

## Implicit formulation of penalty term.

onsider the equation ( Evolution with penalty term ) involving explicit time discretization of first order: The explicit formulation is selected for calculational convenience. At the initial time step , the column satisfies the condition . After evolution for one step, it falls into . Under explicit formulation, it takes at least another time step for the penalty term to take affect and the solution already deviated from . Clearly, we would have to make very small time steps to keep the deviation small. Furthermore, we will pick up discrepancy on every step and accumulate it.

Therefore, we arrive to implicit formulation: The and superscripts over term mark discrepancy of grids. The is constructed with respect to an adaptive basis selection : and , are connected similarly via . The transformation adapts to the grid . Hence, before performing the operation , we apply such transformation and remove grid-dependent details from consideration:

 (Same grid reduction)
where is calculated by projecting final payoff on :

To see that such transformation is correct, put .