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XIV: 09, 102-103, LNM 784 (1980)
MEYER, Paul-André
Sur un résultat de L. Schwartz (Martingale theory)
the following definition of a semimartingale $X$ in a random open set $A$ is due to L. Schwartz (Semimartingales dans les variétés..., Lecture Notes in M. 780): $A$ can be represented as a countable union of random open sets $A_n$, and for each $n$ there exists an ordinary semimartingale $Y_n$ such $X=Y_n$ on $A_n$. It is shown that if $K\subset A$ is a compact optional set, then there exists an ordinary semimartingale $Y$ such that $X=Y$ on $K$
Comment: The results are extended in Meyer-Stricker Stochastic Analysis and Applications, part B, Advances in M. Supplementary Studies, 1981
Keywords: Semimartingales in a random open set
Nature: Exposition, Original additions
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XV: 31, 490-492, LNM 850 (1981)
STRICKER, Christophe
Sur deux questions posées par Schwartz (Stochastic calculus)
Schwartz studied semimartingales in random open sets, and raised two questions: Given a semimartingale $X$ and a random open set $A$, 1) Assume $X$ is increasing in every subinterval of $A$; then is $X$ equal on $A$ to an increasing adapted process on the whole line? 2) Same statement with ``increasing'' replaced by ``continuous''. Schwartz could prove statement 1) assuming $X$ was continuous. It is proved here that 1) is false if $X$ is only cadlag, and that 2) is false in general, though it is true if $A$ is previsible, or only accessible
Keywords: Random sets, Semimartingales in a random open set
Nature: Original
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