| YanJiaAn.[J].数学年刊A辑,1980,1(3-4):545~551 |
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| SOME FORMULAS FOR THE LOCAL TIME OF THE SEMIMARTINGALES |
| Received:January 18, 1980 |
| DOI： |
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| 英文关键词: |
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| 英文摘要: |
| Let X be a semimartingale,we denote by L(X)(resp.\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over L} (X)\]),the local time at 0 of X in the sense of Meyer(resp,Jacod).we establish the following theorems.
Theore 1 Let X be a semimartingale,we have
\[\begin{array}{l}
{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over L} }_t}(X) = \int_0^t {{I_{[{s^ - } = 0]}}} d|{X_s}| - \sum\limits_{0 < s \le t} {{I_{[{s^ - } = 0]}}} |{X_s}|\\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over L} (X) = \frac{1}{2}[L(X) + L( - X)]\L(X) = L({X^ + }),\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over L} (X) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over L} ( - X) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over L} (|X|)
\end{array}\]
Theore 2 Let X and Y be two semimartingale,we have
\[{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over L} }_t}(XY) = \int_0^t {|{X_{{s^ - }}}} |d{{\hat L}_s}(Y) + \int_0^t {|{Y_{{s^ - }}}} |d{{\hat L}_s}(Y)\]
Theore 3 Let X be a semimartingale,and f be a position convex function on R,such that \[f(x) = 0 \Leftrightarrow x = 0\],then we have
\[{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over L} }_t}(f(X)) = {f^'}(0)[\int_0^t {{I_{[{X_{{s^ - }}} = 0]}}} d{X_s} - \sum\limits_{0 < s \le t} {{I_{[{X_{{s^ - }}} = 0]}}} {X_s}] + \frac{1}{2}\rho (\{ 0\} ){{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}
\over L} }_t}(X)\]
where \[{f^'} = \frac{1}{2}({f^'}_{right} + {f^'}_{left})\]and \[\rho \] is the second derivative of f in the sense of distributions
corollary Let X be a semimartingale,we have \[\hat L(|X|) = 0{\kern 1pt} \] for \[\alpha > 1\].If \[\hat L(X) \ne 0\] then for \[0 < \beta < 1\],\[{X^\beta }\]is note a semimartingale.
Theore 4 Let X be a semimartingale,the following statement are equivalent
\[i)X \in (\sum )\hat = (Y = N + V:N{\kern 1pt} \] N is a continuous local martingale,and V is a process of finite variation such that dV is supported by \[\{ s:{Y_{{s^ - }}} = 0\} \} \];
\[ii)\int_0^o {{I_{[{X_{{s^ - }}} \ne 0]}}} d{X_s}\] is a continuous local martingale
\[iii)\int_0^o {{X_{{s^ - }}}} d{X_s}\](or \[{X^2} - [X,X]\]) is a continuous local martingale.
corollary Let X be a continuous semimartingale.then we have \[X \in (\sum ) \Leftrightarrow |X| \in (\sum )\]
let (Ct) be an adapted right continuous process of finite variation,set \[{j_t} = \inf \{ s:{C_s} > t\} ,\bar {\cal F} = {{\cal F}_{jt}},{B_t} = {C_{t - }}\] the following is a variant of Azema-Yor formula.
theorem 5 let X be a semimartingale such that \[{X_{jt}}{I_{[js < \infty ]}} = 0\] a.s,for any \[t \in {R^ + }\] if (\[{h_i}\]) is a bounded (\[{{\bar {\cal F}}_t}\])-predictable process,then (\[{h_{{B_t}}}\]) is (\[{{\cal F}_t}\])-predictable,and we have
\[\begin{array}{l}
\int_0^t {{h_{{B_s}}}} d{X_s} = {h_{{B_s}}}{X_t} - {h_0}{X_0}\\hat L({h_B}X) = \int_0^t | {h_{{B_s}}}|d{{\hat L}_s}(X)
\end{array}\] |
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