Download A Change-of-Variable Formula with Local Time on Curves by Peskir G. PDF

Download A Change-of-Variable Formula with Local Time on Curves by Peskir G. PDF

By Peskir G.

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Extra resources for A Change-of-Variable Formula with Local Time on Curves

Example text

19). 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.

55) 534 Peskir ±ε b±ε (X) t (Z) = t where sign(0) = 0. 56) . 0 Setting Hsε = sign(Zs ) − sign(Zs ± ε) hence we ﬁnd sup | 0 s t b±ε b s (X) − s (X)| t 2ε + 0 t + 0 |Hsε ||µ(s, Xs )| ds |Hsε | dV (b)(s)+ sup 0 s t t 0 Hsε σ (s, Xs ) dBs . a. s in [0, t]. a. s in [0, t]. 57) tends to zero outside a P -null set as ε ↓ 0. 57) tends to zero outside a P -null set as ε ↓ 0. 57) we can make use of the BurkholderDavis-Gundy inequality which yields: s sup E 0 s t 0 Hrε σ (r, Xr ) dBr t E 0 1/2 (Hsε )2 σ 2 (s, Xs ) ds .

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). 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).