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.

## Choosing probing functions.

e take to be scaled and transported hut functions. Let be the hut function and for some fixed scale .

One might think that selection of should be adaptive and similar to selection of the wavelet basis . We argue to the contrary that there is no reason to do such adaptive selection. Indeed, the linear span of includes piecewise linear functions with singularities at 's nodes. Wavelet basis is selected adaptively mostly to compensate for lack of smoothness of approximated solution and payoff function. A system of transported hut functions do not have such difficulty. In fact, if edges of the payoff function are included in the set of nodes of the hut functions then the payoff function is included in linear span of exactly. We only lack precision because we approximate the solution with a piecewise linear functions. But for a multistep penalty function such approximation is acceptable.

The above does not mean, however, that the scale of the probing functions may be crude. Consider a hut function installed at the right boundary of the area where penalty term is in effect. The right half of such hut function will introduce a penalty value into a small area where there should not be any such value. This consideration is the principal reason that prevents selection of crude scale for probing functions. It also follows that the scale of the probing functions must increase with the strength of the penalty term (parameter ).

To compensate, we note that we can calculate an apriori region where the penalty function would be zero. There is no reason to place any probing functions into such region. Such region would be time-dependent and nothing prevents us from selecting probing functions in time-dependent manner.