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.
 A. Example Black equation parameters.
 B. Reduction to system of linear algebraic equations for Black PDE.
 C. Adaptive time step for Black PDE.
 D. Localization.
 E. Reduction to system of linear algebraic equations for q=1.
 F. Preconditioner for Black equation in case q=1.
 G. Summary for Black equation in case q=1.
 H. Implementation of Black equation solution.
 8 Solving one dimensional 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 Black PDE.

e apply technique of the section ( Finite elements ) and results of the section ( Calculation of approximation spaces in one dimension ) to the following problem.

Problem

(Example Black problem) Calculate where is standard Brownian motion and are given positive numbers:

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

Problem

(Example strong Black problem) Find s.t.

We would like to have homogeneous boundary conditions. Hence, we make a change of the unknown function We introduce the notation

Problem

(Example strong Black problem 2) Find s.t.

It is convenient to make a logarithmic change of variable to remove from the equation. We are not going to do it. We are using this problem to test our recipes in preparation for more elaborate problems. For example, affine equations do not have a change of variables that would remove all infinite terms from the PDE. The extreme stiffness that the multiplier brings into the problem is a persistent difficulty when solving financial problems. In the following sections we will completely remove this difficulty while using efficient techniques of generic nature.

The advantages of log-change should be weighted against the cost of decomposing final payoff against wavelet basis. Without the log-change, the payoff is piecewise linear. It may be decomposed almost instantly even in multi-dimensional situation. After log-change it becomes difficult.

 A. Example Black equation parameters.
 B. Reduction to system of linear algebraic equations for Black PDE.
 C. Adaptive time step for Black PDE.
 D. Localization.
 E. Reduction to system of linear algebraic equations for q=1.
 F. Preconditioner for Black equation in case q=1.
 G. Summary for Black equation in case q=1.
 H. Implementation of Black equation solution.