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For a Markov process $X_t$, define a stopping time $\tau$ and $P_x(\cdot)=P(\cdot|X_0=x)$, $$P_x(X_t=x, \tau<t)=P_x(X_t=x|\tau<t)P_x(\tau<t)=P_{X_{\tau}}(X_{t-\tau}=x)\mathbb{E}_x\mathbb{1}_{\tau<t}$$

My question is that how to the last equation.

Is the first term that using the strong Markov property and the second term is $$P_x(\tau<t)=P(\tau<t|X_0=x)=\mathbb{E}(\mathbb{1}_{\tau<t}|X_0=x)?$$

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Showing that $\mathbb{P}_x(\tau<t)=\mathbb{E}_x[1_{\{ \tau<t \}}]$ is just a matter of expanding definitions.

The thing with the first factor is using the strong Markov property to restart the process at time $\tau$.

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