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.
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.
A. Infinitely divisible distributions and Levy-Khintchine formula.
B. Generator of Levy process.
C. Poisson point process.
D. Construction of generic Levy process.
E. Subordinators.
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.

Infinitely divisible distributions and Levy-Khintchine formula.


efinition

(Characteristic exponent of a p.m.) Let $\mu$ be a probability measure on $\QTR{cal}{R}^{n}$ . The characteristic exponent MATH is defined by the relationship MATH

Definition

(Infinitely divisible p.m.) The probability measure $\mu$ on $\QTR{cal}{R}^{n}$ is "infinitely divisible" if for any integer $m$ there exists a p.m. $\mu_{m}$ such that MATH

Proposition

(Levy-Khintchine formula 1) A function MATH is a characteristic function of an infinitely divisible p.m. iff there are MATH , positive semi-definite matrix $n\times n$ matrix $Q$ and a measure $\mu$ on MATH with MATH such that MATH for any MATH .

Equivalent formulation of the above proposition may be obtained if the cut off function MATH is replaced with MATH . This way we have a smooth function with equivalent behavior at $x=0$ and $x=\infty$ .

Proposition

(Levy-Khintchine formula 2) A function MATH is a characteristic function of an infinitely divisible p.m. iff there are MATH , positive semi-definite matrix $n\times n$ matrix $Q$ and a measure $\mu$ on MATH with MATH such that MATH for any MATH .





Notation. Index. Contents.


















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