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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 II. Implementation and results.


he procedure of the previous section ( Mix of backward induction and penalty term approaches II ) is implemented in the script soMix2.py located in the directory OTSProjects/python/wavelet2.


Mix2 figure 1
Plot of $g-z$ .


Mix2 figure 2
Plot of $g-z$ after running the procedure for MATH .


Mix2 figure 3
Plot of $g-z$ after running the procedure for MATH .


Mix2 figure 4
Plot of $g-z$ after running the procedure for MATH .

The slight negative component is introduce by the procedure because the wavelet basis MATH approximates the probing hut functions with limited precision. Hence, the precision may be improved by adding more wavelets in the problem area and/or by using higher order spline function of the section ( Spline functions ) as probing functions.





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