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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
A. Space of distributions. Weak derivative.
B. Fundamental solution.
C. Fundamental solution for the heat equation.
11. Functional Analysis.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Fundamental solution.

et $L$ be a linear differential operator: MATH where MATH and $\phi,a_{k},b$ are functions MATH .


A fundamental solution for the partial derivatives operator $L$ is the distribution $\QTR{cal}{E}$ with the following property MATH

Note that the equation

MATH has the solution MATH where the * is the convolution operation MATH

Indeed, since differentiation commutes with convolution, MATH


(Fundamental solution for ODE). Consider the ODE operator MATH where the $u$ and $a_{k}$ are functions MATH . We conjecture a fundamental solution of the form MATH where the $\theta$ is the step function (see the formula ( step function )) and the $w$ is some unknown smooth function. We substitute into MATH and calculate the derivatives MATH The functions MATH are supported at 0. Hence, if we set MATH then MATH We conclude that the $w$ is defined by the following Cauchy problem MATH

Notation. Index. Contents.

Copyright 2007