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.

## Brownian motion hitting a barrier. ur goal is to compute the probability that the standard Brownian motion hits a given level before time reaches . We are going to use the ( Reflection principle ). First, we note that the condition of the reflection principle is essential. Indeed, suppose we forget about it and write  Therefore, in order to use the reflection principle correctly, we split values into the two intervals:   We perform the change of variable in the second integral.  This result may be understood in line of the proof of the the reflection principle. For every path that hits the level and ends below there is a reflection that hits the level and ends symmetrically above . Hence, one comes directly from those scenarios when the path goes to , and another comes from the inverted paths.

 Notation. Index. Contents.