Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
Printable PDF file
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.

Space of distributions. Weak derivative.


efinition

(Weak derivative). Let $\QTR{cal}{D}$ be a collection of infinitely smooth functions MATH with compact support. We define topology by stating that MATH iff all supports are included in a compact set $S$ and for any $m$ all derivatives of order up to $m$ converge uniformly to 0 on $S$ . We call "distribution on $\QTR{cal}{D}$ " any linear continuous functional on $\QTR{cal}{D}$ . We use the notation MATH The $\QTR{cal}{D}$ is included in MATH ( MATH =space of distributions) by MATH ( MATH acts on MATH according to this rule, thus MATH ). The derivative of the distribution MATH is the distribution MATH that acts according to the rule MATH

One could think at this point that the last definition may be extended to the functions acting MATH for some $a$ and $b$ . Even though such extension is formally possible it would be of little use because the main tools of the following analysis are convolution and Fourier transform.

Example

Consider the MATH given by the step function

MATH (step function)
MATH The derivative $\theta^{\prime}$ is called Dirac's delta function:
MATH (Dirac delta function)
We will use the notation MATH

The Fourier transform and the convolution extend to MATH similarly to the differentiation.





Notation. Index. Contents.


















Copyright 2007