By Peskir G.
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Additional info for A Change-of-Variable Formula with Local Time on Curves
52) where the ﬁnal (strict) inequality follows from the fact that F, LX F, µFx and Fx are locally bounded on C and D. 28) is satisﬁed for all δ > 0. 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.
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.
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. 28) is satisﬁed for all δ > 0. 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).