Abstract
Let $X = \{X_t, t > 0\}$ be a right continuous strong Markov process with state space $E$; let $f$ be a continuous real valued function on $E \times E$; and let $M$ be the time at which the process $\{f(X_{t-}, X_t)\}$ achieves its (last) ultimate minimum. Then conditional on $X_M$ and the value of this minimum, the process $\{X_{M + t}\}$ is Markov and (conditionally) independent of events before $M$.
Citation
P. W. Millar. "A Path Decomposition for Markov Processes." Ann. Probab. 6 (2) 345 - 348, April, 1978. https://doi.org/10.1214/aop/1176995581
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