Use the strong Markov property to show the stationary and independent increments property in Lemma 19.32.c) directly, i.e. for all \(0=u_{0}

Data From Lemma 19.32

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19.32 Lemma. Let (Bt)to be a BM, Lo the local time at zero and Tu its generalized right continuous inverse. a) (Lt)to satisfies Lt+s = LT + Ls 0 = U + LS0T b) (Tu)upo satisfies Tu+v-Tu = Ty Tu c) (Tu)uzo equipped with the filtration = F satisfies for all u, v, 0 E [exp(-(Tu+v - Tu)) | Su] = E [exp(-Tv)] = exp(-v25). (19.33) 4 This simple but fundamental insight is due to It [116]. 5 You should check, that w+ ((w) is indeed measurable, but this follows from the observation that is the first hitting time of zero.