By Peskir G.
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Extra info for A Change-of-Variable Formula with Local Time on Curves
Finance. 15, 169–181. 6. , and Yor, M. (1999). Continuous Martingales and Brownian Motion. Springer, Berlin.
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). 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.
5. 26). 4) as claimed. 49) 0 for 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.
A Change-of-Variable Formula with Local Time on Curves by Peskir G.