Introduction to Stochastic Calculus

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.

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