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.
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.

Outer measure.

et MATH be a measure space (see the definition ( Measurable space )).


(Outer measure) Let $\mu^{\ast}$ be a mapping defined on all subsets of $\Omega$ and valued in MATH . The $\mu^{\ast}$ is called "outer measure" if the following conditions are satisfied:


2. MATH .

3. MATH .


(Measurable set) A set $A$ is called " $\mu^{\ast}$ -measurable" if for any set $B$ we have MATH


$\mu^{\ast}$ -measurable sets constitute a $\sigma$ -algebra. The restriction of $\mu^{\ast}$ to such $\sigma$ -algebra is a complete measure.

Notation. Index. Contents.

Copyright 2007