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.
A. Operations on sets and logical statements.
B. Fundamental inequalities.
C. Function spaces.
D. Measure theory.
a. Complete measure space.
b. Outer measure.
c. Extension of measure from algebra to sigma-algebra.
d. Lebesgue measure.
E. Various types of convergence.
F. Signed measures. Absolutely continuous and singular measures. Radon-Nikodym theorem.
G. Lebesgue differentiation theorem.
H. Fubini theorem.
I. Arzela-Ascoli compactness theorem.
J. Partial ordering and maximal principle.
K. Taylor decomposition.
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.
15. Parabolic PDE.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Complete measure space.


efinition

(Complete measure space) The measure space MATH is "complete" if $\QTR{cal}{F}$ contains all subsets of sets of measure zero.

Definition

(Saturated measure space) The measure space MATH is "saturated" if every locally measurable set is measurable.

Proposition

(Completion of measure via addition of null sets) For a measure space MATH there is a complete measure space MATH such that

1. MATH ,

2. MATH MATH .

3. MATH $A=B\cup N$ where $B\in\QTR{cal}{F}$ and MATH , MATH

Proof

The rule 3 defines a $\sigma$ -algebra. The $\mu$ extends to such $\sigma $ -algebra without ambiguity.





Notation. Index. Contents.


















Copyright 2007