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 I. Basic math.
 1 Conditional probability.
 2 Normal distribution.
 A. Definition of normal variable.
 B. Linear transformation of random variables.
 C. Multivariate normal distribution. Choleski decomposition.
 D. Calculus of normal variables.
 E. Central limit theorem (CLT).
 3 Brownian motion.
 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.

## Calculus of normal variables.

uppose and are two jointly normal random variables: with correlation . We introduce two jointly normal variables and according to the relationships

 (Orthogonal normal variables)
and claim that , are iid . To verify such claim we calculate

Similarly to the relationships ( Orthogonal normal variables ) we may represent any collection of jointly normal variables as a linear combination of iid standard normal variables. We consequently treat the jointly normal variables as vectors in the sense of elementary geometry: In the above calculation, the and act as orthogonal basis and acts like scalar product.

 Notation. Index. Contents.