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Quantitative Analysis
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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.

Operations on sets and logical statements.


e start from an example. We would like to describe the notion of almost sure (=almost everywhere) convergence of the random variables MATH to the random variable $X$ on the set $\Omega$ with respect to a probability $P$ . The English statement would be the following: "There exists a subset $\Omega ^{\ast}$ of the set $\Omega$ such that MATH and for any point $\omega$ of the set $\Omega^{\ast}$ and any however small positive number $\varepsilon$ there exists an integer $N$ depending on $\omega$ and $\varepsilon$ such that for any integer $n$ greater then $N$ the following inequality holds: MATH ." The mathematical notation for such statements is

MATH (Almost sure convergence)
The $\forall$ stands for "any", $\exists$ is "there exists some", s.t. is "such that". Every variable MATH is bound by one of the operators $\forall$ or $\exists$ . If the variable is not bound then the logical statement is meaningless.

Next, we describe a situation when $X_{n}$ does not converge to $X$ on $\Omega^{\ast}.$ Such statement is the exact opposite to the statement above. The general recipe is to permute $\forall$ and $\exists$ and replace the key relationship to the opposite: MATH Again, the boundedness of every variable to $\forall$ or $\exists$ is important.

Let us now describe the notion of convergence in terms of sets. We start from the set of all points $\omega$ where MATH converges to MATH MATH and represent it in terms of more elementary sets: MATH Notably, we describe the $\forall$ ("any") using the intersection and describe the $\exists$ ("exists") using the union of the sets.

Let $\Omega$ be the total set. We would like to describe the equivalent of negation: MATH We note the following property of set operations

MATH (Intersection property)
Hence, MATH We use another property of set operations:

MATH (Union property)
Thus, MATH Note how this result matches the negative statement obtained via the logical operations.

There will be numerous applications of this technique in the present chapter.





Notation. Index. Contents.


















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