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Quantitative Analysis
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Numerical 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.
2. Laws of large numbers.
A. Weak law of large numbers.
B. Convergence of series of random variables.
C. Strong law 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.

Laws of large numbers.


efinition

The r.v. $X$ and $Y$ are called "uncorrelated" if they have finite second moments and MATH . A family of r.v. MATH is "uncorrelated" if $\forall i,j$ , $i\not =j$ , MATH .

Proposition

(Simple law of large numbers) If the family of r.v. MATH is uncorrelated and the second moments have a common bound then MATH in $L^{2}$ , in probability and almost surely.

Proof

We introduce the notation MATH To investigate the convergence in $L^{2}$ , we calculate MATH The MATH are uncorrelated, hence, the second term is zero: MATH Since the second moments have common bound MATH we conclude that MATH Hence, MATH It follows by the proposition ( Convergence in Lp and in probability 1 ) that also MATH

It remains to prove the a.s. convergence. According to the formula ( Chebyshev inequality ) MATH Hence, if we restrict our attention to the subindexing MATH then MATH Therefore, according to the proposition ( Borel-Cantelli lemma, part 1 ), MATH Then, according to the proposition ( IO criteria for AS convergence ), MATH It remains to consider the middle terms MATH for every $n$ . We introduce the notation $D_{n,k}$ : MATH for $k\in$ MATH . We estimate for every $k$ s.t. MATH : MATH

Hence, it suffices to prove that MATH We calculate MATH The MATH are uncorrelated, hence the cross terms vanish: MATH Therefore, according to the formula ( Chebyshev inequality ), MATH It follows, according to the proposition ( Borel-Cantelli lemma, part 1 ), MATH Then, according to the proposition ( IO criteria for AS convergence ), MATH




A. Weak law of large numbers.
B. Convergence of series of random variables.
C. Strong law of large numbers.

Notation. Index. Contents.


















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