Martingale:Fundamentals

• Sample Space

• Event or Outcome

• Intersection and Union

• Random Variable

• Stochastic Process

• Filtration

Let introduce some basic definitions.

Examples:

Lets define a sample space for a coin toss expriment

= { {H}, {T} }

The sample space for the throw of ONE dice is

, can therefore be regarded as a "subset" of the sample space   

Conversely  

can be defined as the set of all outcomes (or events), their intersections and their unions.

The rules of set theory therefore apply.

Lets say we define O as a set of "odd" outcomes of

then the set of even outcomes E has a complement

  implies an "or". In other words the resultant set includes outcomes in 'A' or 'B'

imples an "and". In other words the resultant set includes outcomes in 'A' and 'B'

Example:

Lets consider a set of "odd" outcomes from the above "ONE dice throw" expriment. Let this set be O.

Lets consider a set of "even number" outcomes from the above "ONE dice throw" expriment. Let this set be E.

Let P be a set of prime numbers.

As the figure (1) below shows the results of the intersections and unions of these sets

Figure 1

Random Variable

A random variable is strictly speaking NOT a variable. It is actually a function which assigns a numerical value to  

Figure 2

As shwon in the figure (2) above, the outcomes of the set is mapped to real values by the "function" which we often refer to as a Random Variable

A Stochatic Process S(t) can be viewed as a sequence of random variables indexed by time t.

Hence the stochastic process is a function of both the individual event

and of time t

This can be better illustrated using the Binomial Model.

Let S0 be the value of the asset at time t=0. Let S0move either up by amount U or move down by amount D. So at time t=1, the value of the asset is either US0 or DS0. This processes is illustrated by figure (3) below

Figure 3

As time moves from a discrete process to a continuous process, it becomes difficult to construct the above ever expanding set. This is where the concept of "filtration" comes in handy.

Filtration