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 , 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.