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 be a Levy process in with generator . The space is included in and there are , and a measure on with such that for any we have

Proof

According to the definition ( Levy process ), the increment is infinitely divisible for any . Hence, if is the characteristic exponent of : then According to the proposition ( Levy-Khintchine formula 2 ) thus Therefore, there are , and a measure such that for any , We are aiming to calculate in terms of and . The is connected to the via the relationships By the stationary property of the process, the matrix is only dependent on the distance between and : We introduce the notation thus For any let be a function such that Then Therefore We note that and conclude

 Notation. Index. Contents.