Strict Past Conditioning at Arbitrary Times
Let Open image in new window be a Hunt process with state space (E, ℰ). In  Weil proved the conditional independence of Open image in new window and θT given XT _ for certain hitting times T of the process (Xt _,Xt)t>0. In  Pitman investigated the stochastic dependence of Open image in new window and θT for arbitrary stopping times T. Here we shall look at this question for arbitrary times T. The main result is formula (3) below which contains both Weil’s result, its extension to arbitrary hitting times of (Xt_,Xt) (Theorem 2) and a result for last exit times. This was thought to be an introduction to similar formulae for the excursions straddling arbitrary times, but we had no time to work out the details for the present publication.
KeywordsConditional Independence Random Time Exit Time Arbitrary Time Similar Formula
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