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.
 VI. Scalar product in N-dimensions.
 VII. Wavelet transform of payoff function in N-dimensions.
 VIII. Solving N-dimensional PDEs.

## Penalized problem for mean reverting equation.

he origin of mean reverting equation as stated below was discussed in the section ( Solving one dimensional mean reverting equation ).

Problem

(Mean reverting problem 2) Calculate where the is standard Brownian motion and , , are given positive numbers: and are real numbers.

According to the summary ( Free boundary problem 2 ), the function also solves the following free boundary problem.

Summary

(Mean reverting free boundary problem) We use notation of the problem ( Mean reverting problem 2 ). Let Find s.t.

Let and

Problem

(Mean reverting free boundary problem 2) We use notation of the problem ( Mean reverting problem 2 ). Let Find s.t.

We multiply the above PDE with a smooth function , , integrate over and do integration by parts. We get We add the penalty term according to the recipe ( Variational inequalities in maximization case ). We arrive to the following penalized problem.

Problem

(Penalized mean reverting problem) We use notation of the problem ( Mean reverting problem 2 ). Find a function s.t. where is small parameter and