Adapted Stochastic Processes and Fiber Bundles

Question

Suppose that we have an adapted stochastic process. In other words fix $T>0$ and suppose that for all $t \in [0,T]$ $X(t)$ is a collection of random variables, $F(t)$ is a $\sigma$-algebra, $F(s) \subset F(t)$ when $s<t$, and $X(t)$ is $F(t)$-measurable. Is there a natural way to realize an adapted stochastic process as a fiber bundle?