Consider the following problem: at time $t=0$ a real valued stochastic process $X_{t}$ starts. At some random time $\tau>0$, $Y_{t}$ branches out of $X_{t}$ such that $Y_{0}=X_{\tau}$. After $\tau$ both processes continue to evolve to time $T$. In this case how do we define the filtration $\sigma$-algebras of before and after the random time $\tau$ and for the system as a whole?
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