Correct definition of the probability space of a $ \mathcal{F}_{t} $-measurable random variable

Question

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 ) $

Answer 1

We say that a variable $X_t:\Omega \rightarrow \mathbb{R}$ is $\mathcal{F}_t$-measurable if $X_t^{-1}(A) \in \mathcal{F}_t$ for all $A \in \mathcal{B}(\mathbb{R})$, where $\mathcal{B}$ is the Borel $\sigma$-algebra on the reals.

If by "on the filtered space" you mean adapted to the filtration, then yes, by definition $X_t$ is an $\mathcal{F}_t$-measurable random variable on $(\Omega, \mathcal{F}_t,P)$. The second statement is surely true.

Also, note that $X_t^{-1}(A) \in \mathcal{F}_t \subset \mathcal{F}$ for all $t$ and for all $A\in \mathcal{B}(\mathbb{R})$! Therefore, also the first statement must be correct.