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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.
3. Characteristic function.
A. Basic properties of characteristic function.
B. Convergence theorems for 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.

Basic properties of characteristic function.


roposition

(Uniform continuity of ch.f.) Characteristic function of a r.v. is uniformly continuous in $\QTR{cal}{R}$ .

Proposition

(Ch.f. of a sum) If $X_{1}$ and $X_{2}$ are independent r.v. with d.f. $F_{1}$ and $F_{2}$ then the r.v. $X_{1}+X_{2}$ has a d.f. MATH and ch.f. MATH .

Proof

For the MATH part use the formula ( Total_probability_rule ). The second part is a direct verification.

Proposition

(Inversion of ch.f. into p.m. 1) If $\mu$ is a p.m. induced by a r.v. $X$ then for $x_{1}<x_{2}$ we have MATH

Proof

We proceed by direct verification. We substitute the definition of $f_{X}$ into the claim of the proposition: MATH We would like to invert the order of integration using the proposition ( Fubini theorem ). Hence, we need to verify that the function MATH is MATH integrable over MATH . We estimate MATH and the integral MATH is finite. Hence, the proposition ( Fubini theorem ) is applicable and we reverse the order of integration: MATH We calculate the integral with respect to $t$ : MATH We expand the last integral with the help of MATH and note that MATH is an odd function of $t$ . Hence, MATH . MATH where the last equality holds because MATH is an even function of $t$ . The above is being passed to the limit MATH . Hence, the next task is to calculate the integral of the form MATH First, we calculate the integral MATH The function MATH is sharply decaying a positive infinities for $x$ and $s$ . Hence, the proposition ( Fubini theorem ) applies and we reverse the order of integration: MATH We calculate the internal integral via the repeated integration by parts: MATH MATH Then MATH Note that if $\alpha$ is positive then MATH If $\alpha$ is negative then MATH If $\alpha$ is zero then MATH Hence, MATH

We substitute the last result into the integral MATH Hence, by the proposition ( Dominated convergence theorem ) the limit interchanges with the integral and we recover the result: MATH

Proposition

(Inversion of ch.f. into p.m. 2) If $\mu$ is a p.m. induced by a r.v. $X$ then MATH

Proposition

(Inversion of ch.f. into d.f.). Let $X$ be a r.v. with ch.f. $f_{X}$ and d.f. $F$ . Assume that MATH . Then $F$ is continuously differentiable and MATH





Notation. Index. Contents.


















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