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.

## Stochastic optimization over wavelet basis.

e continue derivations of the sections ( Review of variational inequalities in maximization case ) and ( Penalized problem for mean reverting equation ). We are considering an equation of the problem ( Penalized mean reverting problem ):

The penalty term is non-linear. We will treat it as equation's RHS. We will be manipulating and time step to keep small.

We replace the term with : where is some collection of functions with increasing span when increasing . We will call "probing functions". The set is selection of probing functions, to be discussed later. We discussed at the end of the section ( Impossibility of backward induction ) that

1. It is not possible to replace the scalar product with a projection on a range of functions.

2. There is no need to use a sophisticated collection of probing functions.

This is not a step back from wavelet framework because we make no attempt to project on span of probing functions.

We drop the scale index from notation:

The functions and are given by decomposition and : for a known basis and some index selection . Hence, where the indexes are added to the index selection to point out that these selections may be different. The scalar products are independent of market data and contract parameters and may be precalculated. The memory requirements for storing such data do not increase in multidimensional case because we use tensor product to construct bases.

For calculation purposes we note that where , are Gram matrices and operation is applied to a column component-wise.

Finally ,we remark on inserting the penalty term into the discontinuous Galerkin procedure. We use the summary ( Reduction to system of linear algebraic equations for q=1 ). We insert to maintain correct sign relationship between the penalty term and the time derivative. Then where is some discretization of the integral . The summary ( Summary for mean reverting equation in case q=1 ) applies with the substitution Based on the matrix of the summary ( Summary for mean reverting equation in case q=1 ) we apply the summary ( Summary for Black equation in case q=1, inverted matrix ) to calculate the matrix that transforms

 (Evolution with penalty term)
The following issues need resolution:

1. Selection of the probing functions .

2. Time-discretization of the integral

3. Choosing between implicit and explicit formulations of .

4. Consequences of discontinuity within the definition of .

 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?