By Laporte O.
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Extra info for Series and Ionization Potentials in the Iron Spectrum
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.