I. Basic math.
 1 Conditional probability.
 2 Normal distribution.
 3 Brownian motion.
 A. Definition of standard Brownian motion.
 B. Brownian motion passing through gates.
 C. Reflection principle.
 D. Brownian motion hitting a barrier.
 4 Poisson 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 standard Brownian motion.

et be a random event space (see the section ( Filtration )) and be a standard normal variable (see the section ( Normal variable )) . The standard Brownian motion is a mapping with the following properties:

1. The path is continuous for every .

2. The random variable is independent from (see the section ( Filtration )) for every , .

3. For any , the random variable is distributed as :

 (Brownian motion)

The argument of the Brownian motion is conventionally called "time".

 Notation. Index. Contents.