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


$$
\tau\_a = \sup\{\tau: \mathcal{F}^X\_\tau = \mathcal{F}^Y\_\tau \} \quad\text{ or } \quad
\tau\_a = \inf\{\tau: \mathcal{F}^X\_\tau \neq \mathcal{F}^Y\_\tau \}?
$$



Whether the first moment two stochastic processes differ can be formulated in terms of filtrations?
I think I have found an answer to this question and will post it here for the future reference.


Suppose that we start with two filtrations $\mathbb{H}=(\mathcal{H}\_t)\_{t\ge0}$ and $\mathbb{F}=(\mathcal{F}\_t)\_{t\ge0}$. Consider $\tau\_1$ and $\tau\_2$ - stopping times with respect to both filtrations such that
$$
\mathcal{H}\_{\tau\_1} = \mathcal{F}\_{\tau\_1} \quad \text{ and } \mathcal{H}\_{\tau\_2} = \mathcal{F}\_{\tau\_2}.
$$


Clearly the max of the two ($\tau\_1 \wedge \tau\_2$) is also a stopping time with respect to both filtrations. Moreover,
$$
\mathcal{H}\_{\tau\_1 \wedge \tau\_2} = \sigma(\mathcal{H}\_{\tau\_1}, \mathcal{H}\_{\tau\_2}) = \sigma(\mathcal{F}\_{\tau\_1}, \mathcal{F}\_{\tau\_2}) =\mathcal{F}\_{\tau\_1 \wedge \tau\_2}.
$$


Thus, the set of such stopping times forms a lattice, and we can define
$$
\tau^\* = \sup\{\tau : \mathcal{H}\_{\tau} = \mathcal{F}\_{\tau}, \, \tau \text{ is a stopping time w.r.t. both } \mathbb{H} \text{ and } \mathbb{F} \}
$$
to be the first moment when the two filtrations differ.


