Definition question: Is the expanding history of a stochastic process also a stochastic process but on another probability space?

Question

So usually we take $(\Omega, \mathcal{F}, P)$ and $X_t \in \mathbb{R} \; (\text{event space}) \quad \forall \; t \in \mathbb{R} \;(\text{time})$.

If we consider the process of paths of $X$ up to time $t$, $Y_t \equiv \{X_s: s \leq t\}$, is this still considered a stochastic process on some other extended probability space? Or is this not possible for some technical reason?