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

$(\Omega, \mathcal{F}, \mathcal{P})$

This is a probability triple where
1. $\mathcal{P}$ is the ‘true’ of physical probability measure
2. $\Omega$ is the universe of possible outcomes.
3. $\mathcal{F}$ is the set of possible events where an event is a subset of $\Omega$.

There is also a filtration $\{\mathcal{F}_t\}_{t \ge 0}$, that models the evolution of information through time. For example, if by time $t$, we know that event $\mathcal{E}$ has occurs, then $\mathcal{E} \in \mathcal{F}_t$. In the case of a finite horizon from $[0,T]$, then $\mathcal{F} = \mathcal{F}_T$

A stochastic process $X_t$ is $\mathcal{F}_t$-adapted if the value of $X_t$ is know at time $t$ when the information represented by $\mathcal{F}_t$ is known. Most of the times, we have sufficient information at present.

In the continuous-time model, $\{\mathcal{F}_t\}_{t \ge 0}$ will be the filtration generated by the stochastic processes (usually a brownian motion, $W_t$), based on the model’s specification.

Next, we review some martingales and brownian motion, alongside with quadratic variation here.