Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities

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.
A. Rebalancing wavelet basis.
B. Reduction to system of linear algebraic equations.
7. Solving one dimensional Black PDE.
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.
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Rebalancing wavelet basis.


(Rebalancing wavelet basis) Let $H$ be a Hilbert space, $D$ is a dense set in $H$ and MATH is a basis in $H$ : MATH with the following properties:


2. For any $u\in H$ and any $k$ , MATH

A compact operator $B:H\rightarrow H$ is given by its description on the set $D$ .

We start from a function $u\in H$ , precision $\varepsilon>0$ and an approximation MATH where MATH is a small index subset of $P$ and Proj MATH is projection in $H$ on set MATH .

We would like to find an efficient procedure for calculation of the set MATH or some approximation of $\tilde{K}$ from above. Evaluation of MATH is prohibitively expensive. The matrix MATH is known.

Think of $u_{0}$ being an approximation to a hut function (put straddle). The $u_{0}$ is calculated in the section ( Decomposition of payoff function in one dimension ). We place a lot of high level wavelets around the singularity at the strike. When we evolve $u_{0}$ with Backward Kolmogorov's equation, the singularity disappears. Therefore, the weights $c_{i}$ placed on high order wavelets around the strike should redistribute to lower order wavelets.

We produce MATH by taking MATH with maximal weights $c_{i}$ . Those $w_{i}$ , not included in MATH , would have a small weight $c_{i}$ if included in the sum. When we apply the transformation $B$ , the weights redistribute. The function MATH gives us partial information about such redistribution.

We form the set MATH

We form the set $R_{0}$ recursively via distinct unions MATH The optimal function MATH have to depend on diffusive properties of operator $B$ . However, due to iterative nature of this procedure, MATH would work too. In addition, we add propagation into higher scales as follows:

MATH We denote the construction of $R_{0}$ from $Q_{0}$ as MATH

We calculate MATH Then we calculate MATH MATH We continue MATH MATH for $s=1,2,3,...$ We stop when MATH becomes small.

After completion, the final set $K_{s^{\ast}}$ is reduced by removing indexes with lowest contribution. The result is $\tilde{K}$ .

Another possibility is simply to take $\kappa$ to be a large number and do one iteration.

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Copyright 2007