By Peskir G.
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Additional info for A Change-of-Variable Formula with Local Time on Curves
In particular, this implies that µεn (Aεn ) → µ(A) as n → ∞ outside a P -null set. 45) 0 being true for every t > 0, we see by the previous conclusion that t Ft (s, b(s) + εn ) I (Xs b(s) + εn ) ds 0 t + Fx (s, b(s) + εn ) I (Xs 0 b(s) + εn ) db(s) = µεn (Aεn ) → µ(A). s. as n → ∞. In exactly the same way one proves that: t Ft (s, b(s) − εn ) I (Xs b(s) − εn ) ds 0 t + Fx (s, b(s) − εn ) I (Xs 0 b(s) − εn ) db(s) = νεn (Bεn ) → µ(B). s. 45) (with the minus sign). s. as ε ↓ 0 over a subsequence.
27) hold. 35) note that if x → F (s, x) is convex (concave) then x → Fx (s, x) is increasing (decreasing) so that ε → Fx (s, b(s) ± ε) is increasing (decreasing). 27) by Dini’s theorem (note that each s → Fx (s, b(s) ± ε) is 0 or Fxx continuous on the compact set [0,t]). 34). 26) holds. 28). 37) from c to x with respect to the space variable we ﬁnd that Fx (s, x) − Fx (s, c) = x x x c Fxx (s, y) dy = c G1 (s, y) dy + c G2 (s, y) dy =: H1 (s, x) + H2 (s, x) where ¯ It x → H1 (s, x) is convex and (s, x) → H2 (s, x) is continuous on C¯ and D.
40) above) from Fx to b (X) using integration by parts. 53) for ε > 0. s. as ε ↓ 0 over a subsequence. 40). 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).
A Change-of-Variable Formula with Local Time on Curves by Peskir G.