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 I. Basic math.
 1 Conditional probability.
 A. Definition of conditional probability.
 B. A bomb on a plane.
 C. Dealing a pair in the "hold' em" poker.
 D. Monty-Hall problem.
 E. Two headed coin drawn from a bin of fair coins.
 F. Randomly unfair coin.
 G. Recursive Bayesian calculation.
 H. Birthday problem.
 I. Backward induction.
 J. Conditional expectation. Filtration. Flow of information. Stopping time.
 2 Normal distribution.
 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.

## Definition of conditional probability. et and be two events of the same event space , where is the total event space, is a collection of subsets of equipped with set operations and is a set function acting with the properties for any at most countable collection of sets such that  .

Conditional probability of conditionally on is defined as (Bayes formula)

If the whole space is represented as some disjoint union then for any  (Total probability rule)
The last relationship is called "the total probability rule".

Conditional probability has transitive property: hence, (Transitive bayes formula)

Suppose we are facing calculation of the quantity and the quantity is easily computable. The repeated application of the Bayes formula expresses in terms of . Indeed, the Bayes formula is symmetrical: hence (Inversion remark)

The properties ( Total_probability_rule ),( Transitive_Bayes_formula ) and ( Inversion_remark ) are used repeatedly in the following chapters.

 Notation. Index. Contents.