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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.

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 .

Eisenbaum, N. (2000). Integration with respect to local time. Potential Anal. 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.

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