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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.
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.

Taylor decomposition.


(Taylor decomposition in Peano form) If a function MATH is $n$ times differentiable at $t_{0}$ then MATH


(Taylor decomposition in Schlomilch, Lagrange and Cauchy forms) If a function MATH is $n+1$ times differentiable in MATH , $\delta>0$ then MATH where MATH admits any of the following representations:

(a) MATH

(b) MATH

(c) MATH

for MATH and MATH .


(Integral form of Taylor decomposition) If a function MATH has $n+1$ derivatives in finite interval between $t_{0}$ and $t$ , and MATH is in $L^{1}$ in such interval then MATH


We calculate MATH MATH MATH

Notation. Index. Contents.

Copyright 2007