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.