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.
11. Functional Analysis.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
A. Energy estimates for bilinear form B.
B. Existence of weak solutions for elliptic Dirichlet problem.
C. Elliptic regularity.
D. Maximum principles.
E. Eigenfunctions of symmetric elliptic operator.
F. Green formulas.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Elliptic PDE.


et $U$ be an open bounded subset of $\QTR{cal}{R}^{n}$ . The MATH are functions MATH . The $L$ is a linear differential operator

MATH (Operator L)

Problem

(Elliptic Dirichlet problem). MATH

Definition

(Elliptic differential operator). The operator $L$ is called "elliptic" if the matrix MATH is uniformly positive definite for all $x\in U$ .

Definition

(Bilinear form B). Let MATH , MATH and MATH . We introduce the notation MATH

Let us multiply the equations of the problem ( Elliptic Dirichlet problem ) with a function MATH and integrate with respect to $x$ over $U$ . After integration by parts we obtain MATH

Definition

(Weak solution of elliptic Dirichlet problem). The function MATH is a "weak solution" of the problem ( Elliptic Dirichlet problem ) if it satisfies MATH




A. Energy estimates for bilinear form B.
B. Existence of weak solutions for elliptic Dirichlet problem.
C. Elliptic regularity.
D. Maximum principles.
E. Eigenfunctions of symmetric elliptic operator.
F. Green formulas.

Notation. Index. Contents.


















Copyright 2007