We have seen in Lemma 19.27.a) that (operatorname{supp}left[d L_{t}^{0}(omega) ight] subsetleft{t geqslant 0: B_{t}(omega)=0 ight}) for almost

We have seen in Lemma 19.27.a) that \(\operatorname{supp}\left[d L_{t}^{0}(\omega)\right] \subset\left\{t \geqslant 0: B_{t}(\omega)=0\right\}\) for almost all \(\omega\). Show that \(\operatorname{supp}\left[d L_{t}^{a}(\omega)\right] \subset\left\{t \geqslant 0: B_{t}(\omega)=a\right\}\) for all \(\omega\) from a set \(\Omega_{0}\) with measure 1 such that \(\Omega_{0}\) does not depend on \(a\).

Data From 19.27 Lemma

Transcribed Image Text:

19.27 Lemma. Let (Bt)to be a BM and denote by Lo its local time at zero. a) supp[dL (w)] cZ(w) = {t>0: B(w) = 0} for almost all w. b) Lo~ tlo.