Overview of Stochastic Calculus 2

– P is the “true” or physical probability measure – Ω is the universe of possible outcomes. We use ω ∈ Ω to represent a generic outcome, typically a sample path(s) of a stochastic process(es). – the set F represents the set of possible events where an event is a subset of Ω. • There is also a filtration, {F}t≥0, that models the evolution of information through time. So for example, if it is known by time t whether or not an event, E, has occurred, then we have E ∈ Ft. If we are working with a finite horizon, [0, T ], then we can take F = FT . • We also say that a stochastic process, Xt, is Ft-adapted if the value of Xt is known at time t when the information represented by Ft is known. All the processes we consider will be Ft-adapted so we will not bother to state this in the sequel.