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 I. Basic math.
 1 Conditional probability.
 2 Normal distribution.
 3 Brownian motion.
 4 Poisson process.
 5 Ito integral.
 6 Ito calculus.
 A. Example: exponential of stochastic process.
 B. Example: integral of t_dW.
 C. Example: integral of W_dW.
 D. Example: integral of W_dt.
 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.

## Example: integral of W_dt. e aim to derive distribution of the random variable . According to the formula ( Ito_derivative_of_product ) and rules ( Ito calculus ) We calculate The integral was previously calculated, see the formula ( Int t_dW ): for some normal variable . The variables and are jointly normal (because these are sums of linearly dependent components) with zero mean. It remains to calculate the standard deviation of . We have The only part that we did not consider yet is We introduce a uniform mesh , , , and represent the integral as pre-limit sums with intention to pass back to the limit after some transformations: where the is a collection of independent standard normal variables, . We continue We now pass it back to the integral limit Hence, Notation. Index. Contents.