Let the triple $ \left ( \Omega, \mathcal{F}, \mathbb{P} \right ) $ designates the probability space of a stochastic event and let us consider the filtration $ \left ( \mathcal{F}_{t} \right )_{t \in \mathcal{T}} $, where $ \mathcal{T} = \left \lbrace 1,2,\dotsc, T \right \rbrace $ and $T \in \mathbb{N}$.
Now, let the following sequence of $\mathcal{F}_{t}$-measurable random variables $\left ( X_{t} \right )_{t \in \mathcal{T}} $ be a stochastic process on the filtered probability space $ \left ( \Omega, \mathcal{F}, \left ( \mathcal{F} \right )_{t \in \mathcal{T}}, \mathbb{P} \right ) $.
I was wondering what of the following statements about a specific $X_{t}$ is formally correct:
- $X_t$ is a $\mathcal{F}_{t}$-measurable random variable on $ \left ( \Omega, \mathcal{F}, \mathbb{P} \right ) $
- $X_t$ is a $\mathcal{F}_{t}$-measurable random variable on $ \left ( \Omega, \mathcal{F}_{t}, \mathbb{P} \right ) $