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.

## Mix of backward induction and penalty term approaches I.

e continue research of the previous section. The procedure has several problems:

1. For weak penalty term (high ) the procedure takes several time steps to converge to . For strong penalty term (low ) it overshoots at the spacial boundaries of , taking the solution far beyond desired minimal installation at .

2. It picks up and accumulates discrepancy from correct solution while converging into .

3. Boundary probing functions introduce discrepancy beyond area.

Therefore, we propose to stop advancement in time until we iteratively and minimalistically converge into using collections of internal probing functions of increasing precision.

Without time increment the evolution looks like this: where the vector is input parameter, is a control parameter at our disposal, is a well conditioned matrix and has the form for some wavelet basis and selection of probing functions . We exploit the freedom of selection of probing functions by inserting control parameters : Furthermore, we place probing functions conservatively within the area to make sure that no support of any intersects with . The precision loss that comes from such requirement may be mitigated by increasing scale of the probing functions. The penalty function now takes the form We introduce the notations then Let The control parameters are optimized to achieve with constraints

We perform elementary transformations: and arrive to the problem This is a quadratic optimization problem with linear constraints. To make this evident, we transform the problem to canonical form: The problem takes the form where The gradients for the problem are