Summary
Let X be a strong Markov process. Let M be an optional set with the property that 1MoθT (S)=1 M (s+T) whenever s>0 and T is an optional time with [T]⊂M. If L=sup{t>0∶ tεM}εM, we show that L is a splitting time of X: the pre-L events and the post-L events are conditionally independent given X L . To prove this, we extend work of Sharpe's to show that the big shift operators Θ T and \(\hat \Theta _T\) commute with optional projection and dual optional projection, respectively, whenever T is an optional time. Examples are given which are not contained within previous work of Millar and Getoor.
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Glover, J. Intrinsically homogeneous sets, splitting times, and the big shift. Z. Wahrscheinlichkeitstheorie verw. Gebiete 56, 133–144 (1981). https://doi.org/10.1007/BF00531979
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DOI: https://doi.org/10.1007/BF00531979