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Some remarks on the theory of stochastic integration. (English) Zbl 0737.60047

Séminaire de probabilités, Lect. Notes Math. 1485, 95-107 (1991).
Summary: [For the entire collection see Zbl 0733.00018.]
The main purpose of this article is to propose a reasonable definition for the stochastic integration (S.I.) of progressive processes w.r.t. semimartingales. This S.I. generalizes that of predictable processes w.r.t. semimartingales as well as the stochastic Stieltjes integration. This S.I. is proposed in \(\S 1\). We give also in \(\S 1\) an exponential formula for semimartingales using this S.I. . The rest of this paper consists of several remarks on the theory of stochastic integration which are mostly of pedagogical interest. In \(\S 2\) we propose a new construction of the S.I. of predictable processes w.r.t. local martingales. A simple proof of the integration by parts formula is given in \(\S 3\). Finally, in \(\S 4\) we propose a short proof of Meyer’s theorem on compensated stochastic integrals of local martingales.

MSC:

60H05 Stochastic integrals

Citations:

Zbl 0733.00018
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