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Zbl 0564.60052
Ruiz de Chavez, J.
Sur les intégrales stochastiques multiples. (French)
[A] Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 248-257 (1985).

[For the entire collection see Zbl 0549.00007.] \par The paper is devoted to different methods of construction of multiple integrals. The author gives the general notion of multiple integrals for simple predictable integrands and they may be considered also as iterated integrals. Then he constructs double integrals according to two square integrable martingales as integrators (with restrictions on their characteristics) and according to two special semimartingales. \par Some methods of localisation are considered and compared. Finally, the author gives in elementary form the counter-example of {\it E. Perkins} [see the following review, Zbl 0564.60053], i.e. the example of a sequence of double integrals $\int H\sp n\sb{uv}$ $dM\sb udN\sb v$ where $H\sp n\sb{uv}$ is a simple predictable process (indicator of some stochastic interval), $M\sb t$ and $N\sb t$ are square integrable martingales, $\sum\sb{n}H\sp n\sb{uv}$ converges to a predictable process but $\int H\sp n\sb{uv}$ $dM\sb udN\sb v$ does not converge in probability, so the limit double integral cannot be considered in any usual sense.
[Y.S.Mishu'ra]

MSC 2000:: *60H05 Stochastic integrals

Keywords: multiple integrals; predictable integrands; semimartingales; predictable process

Citations: Zbl 0549.00007; Zbl 0564.60053

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