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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.
VI. Scalar product in N-dimensions.
VII. Wavelet transform of payoff function in N-dimensions.
VIII. Solving N-dimensional PDEs.
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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 MATH where MATH MATH MATH the $W_{t}$ is standard Brownian motion and $T,K,\Delta$ , $A,B,\sigma$ , $\lambda$ are given positive numbers: MATH and $\mu,a$ are real numbers.

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

Summary

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

Let MATH and MATH

Problem

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

We multiply the above PDE with a smooth function $v$ , MATH , integrate over $\left[ A,B\right] $ and do integration by parts. We get MATH MATH 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 MATH s.t. MATH MATH where $\varepsilon>0$ is small parameter and MATH





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