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.
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.
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Localization for mean reverting equation.

imilarly to the section ( Localization ), we are going to calculate the interval $\left[ A,B\right] $ .

In the section ( Mean reverting equation ) we calculated that the SDE MATH may be integrated into MATH We continue MATH The integrals evaluate MATH MATH for a standard normal random variable $\xi$ . We put the results together:

MATH (Mean reverting solution)
where we introduced the convenience notations $\alpha$ and $\beta$ , $\beta>0$ , MATH

Based on a precision $\varepsilon$ , $0<\varepsilon<<1$ , we would like to find two numbers $Y_{-}$ and $Y_{+}$ such that MATH We calculate MATH MATH Let $\xi_{0}$ be a number s.t. MATH then MATH and MATH

The process $Y_{t}$ is connected to $X_{t}$ of the section ( Solving one dimensional mean reverting equation ) by the relationship

MATH (Exponential change)
MATH We aim to set the interval $\left[ A,B\right] $ according to MATH However, we still need to make two more modifications:

1. We need to adapt MATH to the binary structure of our mesh. Similar argument was already made in the section ( Localization ).

2. We want to make $\left[ A,B\right] $ time independent for smaller difference $T-t$ .

To achieve the goal 2 we set MATH MATH We choose $Y_{\pm}$ conservatively wider to preserve precision. Adaptive basis selection is a compensating procedure that preserves efficiency.

For the goal 1, we choose a scale parameter $d_{0}$ and MATH


(Localization for mean reverting equation summary) In context of the problem ( Strong mean reverting problem 2 ) we calculate the interval $\left[ A,B\right] $ according to the following procedure.

Choose an integer scale parameter $d_{0}$ and a precision $\varepsilon$ , $0<\varepsilon<<1$ and initial value $x_{0}$ .

Let $\xi_{0}$ be a number s.t. MATH for standard normal variable $\xi$ .

Calculate MATH where MATH MATH Then MATH

The following is a consequence of the formulas ( Mean reverting solution ) and ( Exponential change ):

MATH (Analytical solution for mean reverting equation)

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Copyright 2007