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I: 06, 72-162, LNM 39 (1967)

**MEYER, Paul-André**

Intégrales stochastiques I--IV (4 talks) (Martingale theory, Stochastic calculus)

This series presents an expanded exposition of the celebrated paper of Kunita-Watanabe (*Nagoya Math. J.* **30**, 1967) on square integrable martingales. The filtration is assumed to be free from fixed times of discontinuity, a restriction lifted in the modern theory. A new feature is the definition of the second increasing process associated with a square integrable martingale (a ``square bracket'' in the modern terminology). In the second talk, stochastic integrals are defined with respect to local martingales (introduced from Ito-Watanabe, *Ann. Inst. Fourier,* **15**, 1965), and the general integration by parts formula is proved. Also a restricted class of semimartingales is defined and an ``Ito formula'' for change of variables is given, different from that of Kunita-Watanabe. The third talk contains the famous Kunita-Watanabe theorem giving the structure of martingale additive functionals of a Hunt process, and a new proof of Lévy's description of the structure of processes with independent increments (in the time homogeneous case). The fourth talk deals mostly with Lévy systems (Motoo-Watanabe, *J. Math. Kyoto Univ.*, **4**, 1965; Watanabe, *Japanese J. Math.*, **36**, 1964)

Comment: This paper was a step in the development of stochastic integration. Practically every detail of it has been reworked since, starting with Doléans-Dade-Meyer 409. Note a few corrections in Meyer 312

Keywords: Square integrable martingales, Angle bracket, Stochastic integrals

Nature: Exposition, Original additions

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Intégrales stochastiques I--IV (4 talks) (Martingale theory, Stochastic calculus)

This series presents an expanded exposition of the celebrated paper of Kunita-Watanabe (

Comment: This paper was a step in the development of stochastic integration. Practically every detail of it has been reworked since, starting with Doléans-Dade-Meyer 409. Note a few corrections in Meyer 312

Keywords: Square integrable martingales, Angle bracket, Stochastic integrals

Nature: Exposition, Original additions

Retrieve article from Numdam