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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
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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.

Measure theory.


efinition

( $\sigma$ -algebra) A collection $\QTR{cal}{F}$ of subsets of a set $\QTR{cal}{\Omega}$ is called a $\sigma$ -algebra if the following conditions hold.

1. MATH .

2. MATH .

3. If MATH , $k=1,...,\infty$ then MATH and MATH .

Definition

(Measurable space) The pair MATH is called a "measurable space" if $\Omega$ is a set and $\QTR{cal}{F}$ is a $\sigma$ -algebra of subsets of $\Omega$ . A set $A$ is called "measurable" if $A\in\QTR{cal}{F}$ . A mapping MATH is called "measure" if MATH and MATH for a countable collection of disjoint sets. The triple MATH is called "measure space". A set $A$ is "locally measurable" if MATH for MATH such that MATH . If, for a measure space MATH , we have MATH then MATH is called "probability space".

Definition

( $\sigma$ -finite set). A set $A\in\QTR{cal}{F}$ is called " $\sigma$ -finite" if it is a countable union of measurable sets of finite measure.




a. Complete measure space.
b. Outer measure.
c. Extension of measure from algebra to sigma-algebra.
d. Lebesgue measure.

Notation. Index. Contents.


















Copyright 2007