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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.
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Mix of backward induction and penalty term approaches I. Implementation and results.


he procedure of the section ( Mix of backward induction and penalty term approaches ) is implemented in the script soMix1.py located in the directory OTSProjects/python/wavelet2.


Payoff figure
The payoff function $g\left( x\right) $ .


Evolution figure
Evolution via mean reverting equation for $\Delta t=0.1$ : MATH . No optimization.


Difference figure
Plot of $z_{0}-g$ .


Corrected difference figure
Plot of MATH for scale of probing hut functions $d=1$ . The $\Omega$ term is localized to left side only. Two probing functions are used.


Corrected difference 2 figure
Plot of MATH for scale of probing hut functions $d=2$ . The $\Omega$ term is localized to left side only. 8 probing functions are used.

We note the following.

(1). The procedure is not effective at the boundary. This is so because we cannot place probing functions at the boundary to avoid overshooting. See the previous section ( Mix of backward induction and penalty term approaches ).

(2). We pick up negative values in the middle of the corrected interval. This happens for two reasons:

(2a). At the edge of the plot of difference $g-z_{0}$ the absolute value of derivative MATH might exceed absolute value of derivative of the probing function.

(2b). The $\Omega$ is a projection of probing function on the wavelet basis MATH . The basis is adapted to $z_{0}$ and $g$ and is not adapted to represent the probing function.

(3). In process of the optimization procedure we have to do considerable number of evaluations of both objective function and constraints.

For these reasons we make yet another modification of the procedure.





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Copyright 2007