XIV: 08, 76-101, LNM 784 (1980)
SHARPE, Michael J.
Local times and singularities of continuous local martingales (
Martingale theory)
This paper studies continuous local martingales $(M_t)$ in the open interval $]0,\infty[$. After recalling a few useful results on local martingales, the author proves that the sample paths a.s., either have a limit (possibly $\pm\infty$) at $t=0$, or oscillate over the whole interval $]-\infty,\infty[$ (this is due to Walsh
1133, but the proof here does not use conformal martingales). Then the quadratic variation and local time of $M$ are defined as random measures which may explode near $0$, and it is shown that non-explosion of the quadratic variation (of the local time) measure characterizes the sample paths which have a finite limit (a limit) at $0$. The results are extended in part to local martingale increment processes, which are shown to be stochastic integrals with respect to true local martingales, of previsible processes which are not integrable near $0$
Comment: See Calais-Genin
1717Keywords: Local times,
Local martingales,
Semimartingales in an open intervalNature: Original
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XIV: 10, 104-111, LNM 784 (1980)
STRICKER, Christophe
Prolongement des semi-martingales (
Stochastic calculus)
The problem consists in characterizing semimartingales on $]0,\infty[$ which can be ``closed at infinity'', and the similar problem at $0$. The criteria are similar to the Vitali-Hahn-Saks theorem and involve convergence in probability of suitable stochastic integrals. The proof rests on a functional analytic result of Maurey-Pisier
Keywords: Semimartingales,
Semimartingales in an open intervalNature: Original Retrieve article from Numdam
