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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.

## Ito calculus. efinition

(Ito calculus). We introduce an operation acting on increments of stochastic processes according to the following rules

1. for a standard Brownian motion .

2. for two independent standard Brownian motions , .

3. .

The operation extends to all Ito processes through the linearity in both arguments:  Suppose is a smooth function of its arguments. The Ito formula states (Ito formula)

The Ito formula is a direct consequence of the Taylor formula and the considerations of the previous section. Indeed, we are always interested in the representation of the difference in terms of the infinitesimal increments: Hence, we apply the Taylor formula to the and keep the leading terms. The term is intended to be used under integral sign. The terms and vanish when we take the integral (pass to the dt=0 limit and take a sum) but the term does not vanish by the formula ( Ito isometry ). This test of survival under the limit dt=0 and sum determines the rules ( Ito calculus ) at the beginning of this section.

For several variables the Ito formula takes form (Ito formula 2)
The following is an important consequence of the Ito formula: (Ito derivative of product)
To see this consider a linear combination of , and .

 A. Example: exponential of stochastic process.
 B. Example: integral of t_dW.
 C. Example: integral of W_dW.
 D. Example: integral of W_dt.
 Notation. Index. Contents.