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| ON LOCAL TIMES AND TRAJECTORIES OF THECONTINUOUS LOCAL MARTINGALES |
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Citation： |
Zheng Weian,He Shengwu.ON LOCAL TIMES AND TRAJECTORIES OF THECONTINUOUS LOCAL MARTINGALES[J].Chinese Annals of Mathematics B,1980,1(3-4):505~510 |
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Net amount： 1268 |
Authors： |
Zheng Weian; He Shengwu |
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| Abstract： |
Let \(M = {({M_t})_{t > 0}}\) be a continuous local martingale, then for every real a, tbere is a decomposition
\[\left| {M - a} \right| = {N^{(a)}} + {L^{(a)}}\]
where \({N^{(a)}}\) is a continuons local martingale and \({L^{(a)}}\) a null-initial-valued continuons predictable increasing process, i. e. local time of M at a.
A point t is called a right (left) a-osoillatory point of a continuous function f
defined on \({R_ + } = [0,\infty )\)，if for all \(\varepsilon > 0\),\(f - a\) is a sign-ohanging function over \((t,t + s)\) \((t-s,t)\). A point t is called a two-sided a-oseillatory point of f, if t is both a right
and a left a-osoiHatory point.
In this paper we have shown:
(1) With probability one measure \(d{L^{(a)}}(\omega )\) concentrates on the set of all two-sided a-osoillatory points of \(M.(\omega )\).
(2) M is uniquely determined by its initial value \({M_0}\) and local times \({L^{(a)}},( - \infty < a < \infty )\). Furthermore M can be constructed with initial value \({M_0}\) and local times \({L^{(a)}},( - \infty < a < \infty )\).
(3) Let T be a stopping time, then for almost all\(\omega \in [T < \infty ]\) either \(T(\omega )\) is a right oscillatory point of \(M.(\omega )\), or there exists \(S(\omega ) > T(\omega )\), such that \(M.(\omega )\)is
constant on \((T(\omega ),S(\omega ))\). |
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