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.

Generator of Levy process.


roposition

(Generator of Levy process) Let $X_{t}$ be a Levy process in $\QTR{cal}{R}$ with generator $L$ . The space MATH is included in $D\left( L\right) $ and there are $a\in\QTR{cal}{R}$ , $\sigma\geq0$ and a measure $\mu$ on MATH with MATH such that for any MATH we have MATH

Proof

According to the definition ( Levy process ), the increment $X_{t+s}-X_{t}$ is infinitely divisible for any $t\geq 0,~s>0$ . Hence, if MATH is the characteristic exponent of $X_{t+s}-X_{t}$ : MATH then MATH According to the proposition ( Levy-Khintchine formula 2 ) MATH thus MATH Therefore, there are $a\in\QTR{cal}{R}$ , $\sigma\geq0$ and a measure $\mu$ such that for any $s>0$ , MATH We are aiming to calculate MATH in terms of $a,\sigma$ and $\mu$ . The $P_{h}$ is connected to the $\Psi_{h}$ via the relationships MATH MATH By the stationary property of the process, the matrix $\pi$ is only dependent on the distance between $x$ and $y$ : MATH We introduce the notation MATH thus MATH For any MATH let $\hat{f}$ be a function such that MATH Then MATH Therefore MATH MATH We note that MATH and conclude MATH





Notation. Index. Contents.


















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