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SOME FORMULAS FOR THE LOCAL TIME OF THE SEMIMARTINGALES

YAN JIAAN(Institute of Mathematica, Academia Sinica)  
Let X be a semimartingale, we denote by L(X) (resp. (X)) the local Sime at O of X in the sense of Meyer(resp. Jacod). We establish the following theorems.Theorem 1. Let X be a semimartingale. We haveTheorem 2. Let X and Y be two semimartingales. We haveTheorem 3. Let X be a semimartingale, and f be a positive convex function on R such that.then we have where and ρ is the second derivative of f in the sense of the distributions.Corollary. Let X be a semimartingale. We have (|X|)=0 for a1. If (X) ≠0, then for 0β1, |X|~β is not a semimartinyale.Theorem 4. Let X be a semimartingale. The followiny statements are equivalent.ⅰ) X∈(Σ) {Y=N+V:N is a continuous local martingale, and V is a process of finite variation such that dV is supported by{s:Y_(s-)=0}};ⅱ)is a continuous local martingale;ⅲ)is a continuous local martingale.Corcllcry. Let X be a continuous semimartinyale. Then we haveLet (C_t) be an adapted right continuous process of finite variation. Set The following is a variant of Azema-Yor formula.Theorem 5. Let X be a semimartingale such that a.s. for any t∈R_+. If (h_t) is bounded ((?)_t)-predictable process, then (h_(B_t)) is (F_t)-Predictable, and we have
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