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 $\Omega$ is the event space. It is the full description of what may happen in the model. By the time moment $t_{1}>0$ 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 $t_{1}$ . Such statements are represented by subsets of $\Omega$ 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 $X_{t}$ described by the picture. The process $X_{t}$ starts at the point A at time $t_{0}$ . By the time $t_{1}$ the process may be observed at point $B_{1}$ or $B_{2}$ . Such outcome is uncertain and these are all possibilities at the time $t_{1}$ . Similarly, if the process is at point $B_{2}$ then it may jump up to $C_{3}$ or down to $C_{2}$ .

We introduce the notation MATH The highlighted path then is described by the elementary random event . At the initial time moment $t_{0}$ our knowledge is given by the trivial algebra with $\Omega$ being enumeration of everything that may happen: MATH and $\emptyset$ being the event that never happens. At the time $t_{1}$ we know where the process went at $t_{0}$ . 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 $t_{2}$ is represented by the algebra . For example, contains the set . The information at $t_{3}$ is represented by the algebra , , , , , , , , .

By definition, a process $Y_{t}$ is adapted to a filtration

if for any particular

the $Y_{t}$ is $\QTR{cal}{F}_{t}-$ measurable. Equivalently, $Y_{t}$ is adapted to

if for any

the algebra $\QTR{cal}{F}_{t}$ contains the full description of the path

for all times up to

. For a particular process $Y_{t}$ one may form a family

such that for any

the algebra $\QTR{cal}{F}_{t}$ is the minimal algebra that makes the random variable $Y_{t}$ $\QTR{cal}{F}_{t}-$ measurable (for any

the $\QTR{cal}{F}_{t}$ is the minimal description of all possible realizations of

). Such a family

is called "the filtration generated by the $Y_{t}$ ". The notation

is commonly used to describe the generated filtration.

Suppose $Y_{t}$ is adapted to

and

The $Y_{s}$ is $\QTR{cal}{F}_{t}$ -measurable (because $\QTR{cal}{F}_{t}$ is sufficient to describe $Y_{t}$ , hence, it contains description of $Y_{s}$ as well). Regularly (unless $Y_{t}$ is deterministic during

), $Y_{t}$ is not $\QTR{cal}{F}_{s}$ -measurable because $\QTR{cal}{F}_{s}$ is lacking structure to describe $Y_{t}$ . However, we may create a crude adjustment of $Y_{t}$ to $\QTR{cal}{F}_{s}$ by taking the conditional expectation

for every set

from $\QTR{cal}{F}_{s}.$ This creates a mapping from $\QTR{cal}{F}_{s}$ to the range of $Y_{t}$ . A proper restriction of such mapping is an $\QTR{cal}{F}_{s}-$ 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 $\QTR{cal}{F}_{s}$ -measurable and satisfy

for any set

from $\QTR{cal}{F}_{s}$ .

The condition MATH

written in the form

Example

In the setting of the previous example, the , , is the random variable taking value if $d_{0}$ and value if $u_{0}$ . 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 $\QTR{cal}{F}_{t}$ if for any smooth function

we have

A random variable $\tau$ is called the $\QTR{cal}{F}_{t}$ -stopping time if for any

the random event

belongs to the $\QTR{cal}{F}_{t}$ . In other words, $\tau$ 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 $X_{t}$ in our examples, the time moment first time when the $X_{t}$ is above level $L\}$ is a stopping time. The time moment the time when $X_{t}$ reaches maximum for is not a stopping time.