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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.

Real Variable.


n the chapter ( Conditional probability chapter ) we introduced notions of random events and illustrated how statements about realization of random variables could be treated as operations on sets and measured via probability function defined on these sets. In the present chapter we expand such approach.

The references for this chapter are [Royden] , [Chung] and [Kolmogorov] .

The triple MATH is called "probability space". Here the $\Omega$ is an event space, $\QTR{cal}{F}$ is a $\sigma $ -algebra of subsets of $\Omega$ (see the section ( Operations on sets and logical statements )) and $P$ is a probability measure (p.m.) MATH , MATH and $P$ is $\sigma$ -additive. We denote MATH . MATH is the Borel field with MATH and MATH . "Borel field $\QTR{cal}{B}$ " is a minimal $\sigma$ -algebra containing all open sets.

Definition

(Real valued random variable). Let MATH . The mapping MATH is called (" $\QTR{cal}{F}$ -measurable") "random variable" (r.v.) if for any MATH MATH

Definition

(Borel measurable function) The mapping MATH is a "Borel measurable function" iff MATH

The random variable $X$ on MATH induces the triple MATH according to the rule MATH For a measure MATH we introduce a (cumulative) distribution function (d.f.) MATH , MATH




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.

Notation. Index. Contents.


















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