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.

Construction of generic Levy process.


(Construction of generic Levy process) Let MATH , MATH and $\mu$ is a measure on MATH such that MATH . Then there exists a filtered probability space MATH and a Levy process $X_{t}$ such that MATH


Let $W_{t}$ be a standard Brownian motion in $\QTR{cal}{R}^{n}$ and $e_{t}$ is an independent from $W_{t}$ Poisson point process in $\QTR{cal}{R}^{n}$ (see the definition ( Poisson point process )) with characteristic measure $\mu$ and $\delta=0$ . We introduce the processes MATH The family MATH has an MATH limit with respect to the norm MATH for any $t>0$ . The convergence in MATH insures that the limit is a Levy process.

The processes MATH have the characteristic exponents MATH : MATH Thus MATH is a sum of the independent processes and has the characteristic exponent MATH as claimed.

We calculate the characteristic functions as follows. MATH To calculate MATH we note the definition ( Characteristic measure of Poisson point process ): MATH Hence, for small $h$ we use the definition ( Poisson point process )-2,5: MATH Thus MATH The above means that the probability that $e_{s}\in A$ twice within a small time interval is negligible. Similarly, MATH For the point $\delta=0$ (see the definition ( Poisson point process )) we have MATH The rest of the calculation follows the technique developed in the chapter ( Poisson process ). For a fixed $N$ we set $h=\frac{1}{N}$ , $t_{k}=kh$ , MATH MATH Note that MATH thus MATH and we continue MATH Note that the last expression is of the form MATH we perform the Taylor expansion of log: MATH Hence MATH where the $o\left( 1\right) $ term disappeared because $N$ can be of any value and we let $N\rightarrow\infty$ .

The expectation MATH is evaluated similarly.

Notation. Index. Contents.

Copyright 2007