Sur les intégrales stochastiques multiples. (French) Zbl 0564.60052

Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 248-257 (1985).
[For the entire collection see Zbl 0549.00007.]
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.
Some methods of localisation are considered and compared. Finally, the author gives in elementary form the counter-example of E. Perkins [see the following review, Zbl 0564.60053], i.e. the example of a sequence of double integrals \(\int H^ n_{uv}\) \(dM_ udN_ v\) where \(H^ n_{uv}\) is a simple predictable process (indicator of some stochastic interval), \(M_ t\) and \(N_ t\) are square integrable martingales, \(\sum_{n}H^ n_{uv}\) converges to a predictable process but \(\int H^ n_{uv}\) \(dM_ udN_ v\) does not converge in probability, so the limit double integral cannot be considered in any usual sense.
Reviewer: Y.S.Mishura


60H05 Stochastic integrals
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