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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.
a. Cauchy and Young inequalities.
b. Cauchy inequality for scalar product.
c. Holder inequality.
d. Lp interpolation.
e. Chebyshev inequality.
f. Lyapunov inequality.
g. Jensen inequality.
h. Estimate of mean by probability series.
i. Gronwall inequality.
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.

Holder inequality.


Let $p,q$ are real numbers $1\leq p,q<\infty$ connected by the relationship MATH and $u\in L^{p}$ , $v\in L^{q}$ . Then

MATH (Holder inequality)


The formula ( Young inequality ) implies MATH Set MATH , MATH where MATH and MATH then MATH Hence, MATH


Assume that MATH is a finite collection of real numbers connected by the relationship MATH and $u_{i}\in L^{p_{i}}$ . Then

MATH (Holder inequality 2)


The proposition is a consequence of the formula ( Holder inequality ) by induction.


Let $\mu$ be a positive function. Assuming that all the integrals are well defined and $p,q$ are connected as above, the following estimate holds

MATH (Holder inequality 3)


We essentially repeat the proof of the formula ( Holder inequality ). MATH Set MATH , MATH where MATH and MATH then MATH Hence, MATH

Notation. Index. Contents.

Copyright 2007