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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.
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.

Fubini theorem.


efinition

The measure space MATH is called "complete" if MATH , MATH .

Proposition

Let MATH and MATH are two complete measure spaces. Then there exists a complete measure space MATH such that MATH and MATH

Proposition

(Fubini theorem) Let MATH and MATH are two complete measure spaces and the space MATH is their product as in the previous proposition. Let $f$ be a MATH -integrable function. Then

1. Almost surely in $x_{1}\in\Omega_{1}$ the function MATH is an MATH -integrable function of $x_{2}$ .

2. Almost surely in $x_{2}\in\Omega_{2}$ the function MATH is an MATH -integrable function of $x_{1}$ .

3. MATH is an MATH -integrable function of $x_{2}$ .

4. MATH is an MATH -integrable function of $x_{1}$ .

5. MATH MATH .





Notation. Index. Contents.


















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