I am, struggling to see whether the first moment when two processes are different (in terms of their finite dimensional distributions) can be defined in terms of their filtrations and would appreciate any suggestions / clarifications.

Here is an example of what I am trying to do: suppose that $X$ is a standard Brownian motion and $Y$ is a stopped BM, i.e. $Y_t = X_{\tau_a \wedge t}$ where $$ \tau_a = \inf\{t > 0 : X_t \ge a\}. $$

Clearly $Law(X_{\tau_a \wedge t}, t \ge 0 ) = Law(Y_{\tau_a \wedge t}, t \ge 0 )$ and $\mathcal{F}^X_{\tau_a} = \mathcal{F}^Y_{\tau_a}$, moreover, $\tau_a$ is in some sense a maximal stopping time $\tau$ such that the stopped processes $(X_{\tau \wedge t})_{t\ge0}$ and $(Y_{\tau \wedge t})_{t\ge0}$ are identical.

Is there a way to recover $\tau_a$ by looking only at the filtrations $\mathbb{F}^X$ and $\mathbb{F}^Y$? In other words is there a way to define $\tau_a$ as

$$ \tau_a = \sup\{\tau: \mathcal{F}^X_\tau = \mathcal{F}^Y_\tau \} \quad\text{ or } \quad \tau_a = \inf\{\tau: \mathcal{F}^X_\tau \neq \mathcal{F}^Y_\tau \}? $$