Let introduce some basic definitions. - Sample Space - it is set of all possible events and this space is denoted by
- Event - the result of an experiment is defined as an event and denoted by
Examples: Lets define a sample space for a coin toss expriment = { {H}, {T} }
The sample space for the throw of ONE dice is = { 1, 2, 3, 4, 5, 6 }
An
event or an outcome, , 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'
Lets consider a set of "even number" outcomes from the above "ONE dice throw" expriment. Let this set be and 'B'Example: Lets consider a set of "odd" outcomes from the above "ONE dice throw" expriment. Let this set be O.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 each event , written as
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 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 S _{0} be the value of the asset at time t=0. Let S_{0}move either up by amount U or move down by amount D. So at time t=1, the value of the asset is either US_{0} or DS_{0}. This processes is illustrated by figure (3) belowAs 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 |