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.

## Conditional expectation. Filtration. Flow of information. Stopping time.

e are considering development of some market model during a time interval At the time we know nothing about the future and we represent this fact with the trivial algebra The is the event space. It is the full description of what may happen in the model. By the time moment random variables of the model have certain realizations. One may make particular statements about such realizations that are perfectly verifiable from the point of view of information available at time . Such statements are represented by subsets of and constitute an algebra , (see section ( Operations on sets ) for explanation of the "algebra" term in this context). Similarly, for a time moment we form an algebra . Since market participants do not forget information, is a subset of . A family of such algebras is called "filtration" or "flow of information".

Example

Consider a discrete random walk described by the picture. The process starts at the point A at time . By the time the process may be observed at point or . Such outcome is uncertain and these are all possibilities at the time . Similarly, if the process is at point then it may jump up to or down to .

We introduce the notation The highlighted path then is described by the elementary random event . At the initial time moment our knowledge is given by the trivial algebra with being enumeration of everything that may happen: and being the event that never happens. At the time we know where the process went at . Hence, where the means "take all intersections, unions and complements of the arguments and put them together into the algebra ". For example, contains the set The information at is represented by the algebra . For example, contains the set . The information at is represented by the algebra , , , , , , , , .

By definition, a process is adapted to a filtration if for any particular the is measurable. Equivalently, is adapted to if for any the algebra contains the full description of the path for all times up to . For a particular process one may form a family such that for any the algebra is the minimal algebra that makes the random variable measurable (for any the is the minimal description of all possible realizations of ). Such a family is called "the filtration generated by the ". The notation is commonly used to describe the generated filtration.

Suppose is adapted to and The is -measurable (because is sufficient to describe , hence, it contains description of as well). Regularly (unless is deterministic during ), is not -measurable because is lacking structure to describe . However, we may create a crude adjustment of to by taking the conditional expectation for every set from This creates a mapping from to the range of . A proper restriction of such mapping is an measurable random variable that we denote The same (in "almost sure" sense) object could be introduced by requiring that the variable by definition, would be -measurable and satisfy for any set from .

The condition written in the form

 (Chain rule)
is called "the chain rule".

Example

In the setting of the previous example, the , , is the random variable taking value if and value if . We would, of course, have to assign some probabilities to the events and to actually calculate these numbers. Observe that the chain rule ( Chain_rule ) is simply a grouping of summation terms in such situation.

We will say that the random variable is independent from if for any smooth function we have .

A random variable is called the -stopping time if for any the random event belongs to the . In other words, is a stopping time if at any moment we are able to tell with certainty which of the statements and is true.

Example

For the process in our examples, the time moment first time when the is above level is a stopping time. The time moment the time when reaches maximum for is not a stopping time.

 Notation. Index. Contents.