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 be a probability measure on . The characteristic exponent is defined by the relationship

Definition

(Infinitely divisible p.m.) The probability measure on is "infinitely divisible" if for any integer there exists a p.m. such that

Proposition

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

Equivalent formulation of the above proposition may be obtained if the cut off function is replaced with . This way we have a smooth function with equivalent behavior at and .

Proposition

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

 Notation. Index. Contents.