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 passing through gates.

ur goal is to calculate the quantity for the standard Brownian motion , some intervals and time moments , . The is probability.

We split the interval into a disjoint union of intervals , . We calculate We use the formula ( Bayes formula ). We make small. We use the formula ( Brownian motion ). We make the change , .

 Notation. Index. Contents.