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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.

Levy process.


efinition

(Stationary independent increments) A process $X_{t}$ is said to have "stationary independent increments" if $X_{t}-X_{s}$ has a distribution depending only on $t-s$ .

The notion of stationary independence is more restrictive the homogeneity of increments because homogeneity only states the manner of time dependence and leaves the possibility of $X_{t}$ -dependence.

Definition

(Levy process) A Feller process with stationary independent increments is called "Levy process". Equivalently, $X_{t}$ is a "Levy process" if for any $t,s>0$ the increment $X_{t+s}-X_{t}$ is independent from MATH and has the same distribution as $X_{s}-X_{0}$ .




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.

Notation. Index. Contents.


















Copyright 2007