Is there a continuous-time analog to the switching principle for Martingales? I'm referring to this: Martingale: show switching of two supermartingales with a stopping time is also a supermartingale.
That is, suppose $X_t^1$ and $X_t^2$ are martingales, $\tau$ is a stopping time, and $X_\tau^1 = X_\tau^2$. Let
$$ Y_t = X_t^1 \mathbf{1}(\tau > t) + X_t^2 \mathbf{1}(\tau \le t). $$
Is $Y_t$ a martingale? I tried to extend the discrete-time proof, but it was not obvious how to do so. If the above is not true in general, are there conditions under which it is true?