Assume that the processes in Lemma 5.7 or in Lemma 5.8 are only "almost surely right continuous", resp. "almost surely continuous". Identify all steps in the proofs where this becomes relevant and show that the assertions remain valid if the filtrations \(\mathscr{F}_{t}^{X}\) and \(\mathscr{F}_{t+}^{X}\) are completed, so that they contain all subsets of measurable null sets.

Data From Lemma 5.7

Data From Lemma 5.8

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5.7 Lemma. Let (Xt)to be a d-dimensional stochastic process with exclusively right continuous sample paths and Uc Rd an open set. The first hitting time T satisfies {Ty 1 (5.9) For the first assertion we claim that for all t > 0 {Tu