Back in undergraduate days, when I took my first module on financial mathematics, my professor introduced us by that the most important things are the following
This is a probability triple where
1. 
is the ‘true’ of physical probability measure
2. 
is the universe of possible outcomes.
3. 
is the set of possible events where an event is a subset of .
There is also a filtration 
, that models the evolution of information through time. For example, if by time 
, we know that event 
has occurs, then 
. In the case of a finite horizon from ![[0,T]](http://theculture.sg/wp-content/ql-cache/quicklatex.com-4378aa1d2e51e4ece82a1b25bbbe624a_l3.png "Rendered by QuickLaTeX.com")
, then 
A stochastic process 
is 
-adapted if the value of is know at time when the information represented by is known. Most of the times, we have sufficient information at present.
In the continuous-time model, will be the filtration generated by the stochastic processes (usually a brownian motion, 
), based on the model’s specification.
Next, we review some martingales and brownian motion, alongside with quadratic variation here.
