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Additional resources for A Change-of-Variable Formula with Local Time on Curves
54). 38) is assumed to be satisﬁed, by Helly’s theorem it w follows that ds Fx (s, b(s) ± ε)→ ds Fx (s, b(s)±) on [0, t] in the sense that t t 0 g(s) ds Fx (s, b(s) ± ε) → 0 g(s) ds Fx (s, b(s)±) as ε ↓ 0 for every continuous function g : [0, t] → R. 54) it is therefore sufﬁcient to b show that outside a P -null set b±ε s (X) → s (X) uniformly over s in [0, t] as ε ↓ 0 (possibly over a subsequence). For this, apply the Tanaka formula to the semimartingale Z = X − b and the function f (z) = |z ± ε| for z ∈ R and ε 0.
28) is satisﬁed for all δ > 0. 28) as claimed. 31) holds then s → F (s, b(s) ± ε) is decreasing on [0, t] and therefore of bounded variation. 28) follows as well. 28) as claimed. 28) follows in the same way. 28) as claimed. 6. 49) above. 50) 0<ε<δ 0 where the ﬁnal (strict) inequality follows from the fact that F and Fx are locally bounded on C and D. 28) is satisﬁed for all δ > 0. 27) holds. 27), respectively. 51) 0 for ε > 0. 52) where the ﬁnal (strict) inequality follows from the fact that F, LX F, µFx and Fx are locally bounded on C and D.