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.

## Removing stiffness from penalized equation. e continue considerations of the previous section.

We are solving a slightly non-linear system of algebraic equations using a Newton technique and we are having problems because of limitations on area of convergence. The next most effective relevant numerical technique is Runge-Kutta technique. In order to apply it, perhaps in combination with the Newton procedure, we need a continuous parameter that would produce a Cauchy problem with smooth RHS. The idea is to make a Runge-Kutta step, then improve using Newton and then do Runge-Kutta again and so fourth.

The system of equations is with The continuous parameter that we need for our Cauchy problem is reversed time. Let Then and solves the equation for . We rewrite the equation: and differentiate with respect to : or and we obtain a Cauchy problem. We now have the function that solves both Cauchy problem and algebraic problem for all . We apply Runge-Kutta and Newton interchangeably, as discussed above.

Next, we calculate the components.

The term was considered in the previous section.

According to the summary ( Summary for mean reverting equation in case q=1 ), Thus However, such approach requires calculation of for variety of . Instead, we do exponentiation. We select a , a power and calculate This way for reasonably large . Such has multiplicative property which is especially useful in present situation. We have The procedure is implemented in the script soCauchy2.py within the directory OTSProjects/python/wavelet2. We are able to make reasonably small time steps without stability and divergence issues. However, the procedure exhibits uncomfortable level of sensitivity to the control parameters , and scale of probing functions. In the following section we discuss reasons of such sensitivity and propose an improved procedure.