I. Wavelet calculations.
 II. Calculation of approximation spaces in one dimension.
 III. Calculation of approximation spaces in one dimension II.
 IV. One dimensional problems.
 1 Decomposition of payoff function in one dimension. Adaptive multiscaled approximation.
 2 Constructing wavelet basis with Dirichlet boundary conditions.
 3 Accelerated calculation of Gram matrix.
 4 Adapting wavelet basis to arbitrary interval.
 5 Solving one dimensional elliptic PDEs.
 6 Discontinuous Galerkin technique II.
 7 Solving one dimensional Black PDE.
 8 Solving one dimensional mean reverting equation.
 A. Reduction to system of linear algebraic equations for mean reverting equation.
 B. Evaluating matrix R.
 C. Localization for mean reverting equation.
 D. Implementation for mean reverting equation.
 V. Stochastic optimization in one dimension.
 VI. Scalar product in N-dimensions.
 VII. Wavelet transform of payoff function in N-dimensions.
 VIII. Solving N-dimensional PDEs.

## Solving one dimensional mean reverting equation.

e study the mean reverting equation in preparation to an experiment in stochastic optimization. The mean reverting equation was introduced in the section ( Mean reverting equation ). We make all parameters constant. A quantity is given by an equation where , . The has the meaning where the quantity is a value of some observable market parameter that enters into a piecewise linear payoff function. We calculate an equation for :

Problem

(Mean reverting problem) Calculate where where is standard Brownian motion and , , are given positive numbers: and are real numbers.

The function of such shape is called "butterfly" payoff. One reason for choosing such payoff becomes apparent right away.

According to the section ( Backward Kolmogorov equation ), the function is also a solution of the following problem.

Problem

(Strong mean reverting problem) Find s.t.

We transform to a problem with homogeneous boundary conditions. Let and

Problem

(Strong mean reverting problem 2) Find s.t.

 A. Reduction to system of linear algebraic equations for mean reverting equation.
 B. Evaluating matrix R.
 C. Localization for mean reverting equation.
 D. Implementation for mean reverting equation.