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 I. Basic math.
 1 Conditional probability.
 2 Normal distribution.
 3 Brownian motion.
 4 Poisson process.
 A. Definition of Poisson process.
 B. Distribution of Poisson process.
 C. Poisson stopping time.
 D. Arrival of k-th Poisson jump. Gamma distribution.
 E. Cox process.
 5 Ito integral.
 6 Ito calculus.
 7 Change of measure.
 8 Girsanov's theorem.
 9 Forward Kolmogorov's equation.
 10 Backward Kolmogorov's equation.
 11 Optimal control, Bellman equation, Dynamic programming.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Definition of Poisson process.

efinition

Poisson process is a piecewise continuous non decreasing random process that takes values in consecutive positive integers. It has the following properties for any : In addition, for any , is independent from and for any , is distributed as . At the jump times we define the process to be continuous on the right.

According to the definition, for some number and infinitesimal . The number is called "intensity" of Poisson process.

In the applications of the following chapters we will be interested in the time of the first Poisson jump. We introduce the notation for the filtration generated by the process . The following properties directly follow from the definitions:

 (Poisson property 1)

 (Poisson property 1a)

 Notation. Index. Contents.