Show that in Lemma 19.32.a) the following stronger assertion holds: (left(L_{t} ight)_{t geqslant 0}) is an additive

Show that in Lemma 19.32.a) the following stronger assertion holds: \(\left(L_{t}\right)_{t \geqslant 0}\) is an additive functional, i.e. \(L_{t+s}=L_{t}+L_{s} \circ \theta_{t}\) holds for all \(s, t\geqslant 0\).

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 b) (Tu)uo satisfies Tu+v - Tu = Ty 0 = U + LS0T Tu c) (Tu)uzo equipped with the filtration S = F satisfies for all u, v, 0 E [exp(-(Tu+v - Tu)) | Su] = E [exp(-Tv)] = exp(-v2) (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.