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.

## Review of variational inequalities in maximization case.

he chapter ( Variational inequalities ) contains a recipe for stochastic minimization of a cost function. Such was a convention of the original source. In financial problems we would like to maximize income. Hence, we briefly review the logic to trace changes to and plus to minus.

We do calculation in context and with notation of the section ( Optimal stopping time problem ). The definitions of , , , , are exactly the same. We introduce differently: We have the same calculation leading to the equation inside the non-stopping area. Exactly one of the equalities or holds at all times. Since we are maximizing the payoff, we would not stop unless Thus,

If the stopping does not occur then thus and

Summary

(Free boundary problem 2 summary) The function satisfies on the conditions where exactly one of the inequalities is strict at all times, thus

Let then

 (Free boundary problem 2)

The weak formulation for the equation is where the minus comes from integration by parts and the expression for is given in the definition ( Bilinear form B 2 ).

At the beginning of the chapter ( Variational inequalities ) we remarked that we need variational inequalities to conduct numerical stochastic optimization. Later within that chapter, in the section ( Penalized evolutionary problem ), we introduce a practical recipe for a minimization case: Precisely, we solve the problem ( Strong formulation of evolutionary problem ) using -limit of the problem ( Evolutionary penalized problem ). Note carefully the role of the penalty term in the equation If is greater than then the penalty term comes into life: This is a backward equation. Hence, positive penalty term means positive time derivative and decreasing solution when going backward in time. The penalty term pushes the solution to the desired property

We are presently considering a maximization case, see formula . We need to keep the solution in the area where Following the results of the section ( Evolutionary variational inequalities ), we insert the penalty term that comes into effect when the opposite happens: . Thus, the penalty term is It needs to push the solution upward when going backward in time. Hence, the derivative should get negative component:

Summary

(Variational inequalities in maximization case) In context of the section ( Variational inequalities ), the function may be evaluated by using recipes of the section ( Evolutionary variational inequalities ) modified by the substitution

In particular, the time derivative and penalty term should match signs according to the rule