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.

## Poisson point process.

efinition

(Poisson point process) Let be a filtered probability space, be a measurable space and , . A -valued process is called "Poisson point process" if it has the following properties.

1. The map is - measurable.

2. The set is countable a.s.

3. There is a sequence of sets s.t. and where

4. The process is -adapted.

5. For any , and any the distribution of conditioned on is the same as the distribution of .

Proposition

(Independence of Poisson processes) Two Poisson processes adapted to the same filtration are independent iff they almost never jump simultaneously.

Proposition

(Properties of Poisson point process) Let be a Poisson point process.

1. is a Poisson process for any .

2. For s.t. the processes and are independent.

3. The quantity is independent of and constitutes a -finite measure on .

Definition

(Characteristic measure of Poisson point process) The measure defined in the proposition ( Properties of Poisson point process ) is called "characteristic measure of Poisson point process".

An example of calculation with Poisson point process may be found within the proof of the proposition ( Construction of generic Levy process ).

 Notation. Index. Contents.
 Copyright 2007