By Peskir G.
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Extra info for A Change-of-Variable Formula with Local Time on Curves
13, 303–328. 2. , and Shiryayev, A. N. (1995). Quadratic covariation and an extenˆ formula. Bernoulli 1, 149–169. sion of Ito’s 3. Kranktz, S. , and Parks, H. R. (2002). The Implicit Function Theorem. Birkh¨auser, Boston. 4. Pedersen, J. , and Peskir, G. (2002). On nonlinear integral equations arising in problems of optimal stopping. Proc. Funct. Anal. VII (Dubrovnik 2001), Various Publ. Ser. No. 46, 159–175. 5. Peskir G. (2005). On the American option problem. Math. Finance. 15, 169–181. 6. , and Yor, M.
In exactly the same way one ﬁnds that µ1,2 (∂A) = 0 outside a P -null set. ± ± Denoting µ± n =µεn for all n 1 we claim that limn→∞ µn (An )=µ1,2 (A) ± outside a P -null set. To see this set anm = µn (Am ) and note that the two limits an∞ = limm→∞ anm = µ± n (A) and a∞m = limn→∞ anm = µ1,2 (Am ) exist (the latter outside a P -null set by the weak convergence established) and moreover satisfy limn→∞ an∞ = limm→∞ a∞m = µ1,2 (A) =: a∞∞ (the former outside a P -null set by the weak convergence established).
43). 28). 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.
A Change-of-Variable Formula with Local Time on Curves by Peskir G.