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XI: 26, 390-410, LNM 581 (1977)

**JACOD, Jean**

Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales (Martingale theory)

This paper develops the theory of stochastic integration (previsible and optional) with respect to local martingales starting from the particular case of continuous local martingales, and from the explicit description of the jumps of a local martingale (1121, 1129). Then the theory of stable subspaces of $H^1$ (instead of the usual $H^2$) is developed, as well as the stochastic integral with respect to a random measure. A characterization is given of the jump process of a semimartingale. Then previsible stochastic integrals for semimartingales are given a maximal extension, and optional integrals for semimartingales (differing as usual from those for martingales) are defined

Comment: On the maximal extension of the stochastic integral $H{\cdot}X$ with $H$ previsible, see also Jacod,*Calcul stochastique et problèmes de martingales,* Springer 1979. Other, equivalent, definitions are given in 1415, 1417, 1424 and 1530

Keywords: Stochastic integrals, Optional stochastic integrals, Random measures, Semimartingales

Nature: Original

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Sur la construction des intégrales stochastiques et les sous-espaces stables de martingales (Martingale theory)

This paper develops the theory of stochastic integration (previsible and optional) with respect to local martingales starting from the particular case of continuous local martingales, and from the explicit description of the jumps of a local martingale (1121, 1129). Then the theory of stable subspaces of $H^1$ (instead of the usual $H^2$) is developed, as well as the stochastic integral with respect to a random measure. A characterization is given of the jump process of a semimartingale. Then previsible stochastic integrals for semimartingales are given a maximal extension, and optional integrals for semimartingales (differing as usual from those for martingales) are defined

Comment: On the maximal extension of the stochastic integral $H{\cdot}X$ with $H$ previsible, see also Jacod,

Keywords: Stochastic integrals, Optional stochastic integrals, Random measures, Semimartingales

Nature: Original

Retrieve article from Numdam