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XIV: 15, 128-139, LNM 784 (1980)
CHOU, Ching Sung; MEYER, Paul-André; STRICKER, Christophe
Sur l'intégrale stochastique de processus prévisibles non bornés (Stochastic calculus)
The standard theory of stochastic integration deals with locally bounded previsible processes. The natural definition of the stochastic integral $H.X$ of a previsible process $H$ w.r.t. a semimartingale $X$ consists in assuming the existence of some decomposition $X=M+A$ such that $H.M$ exists in the martingale sense, and $H.A$ in the Stieltjes sense, and then defining $H.X$ as their sum. This turns out to be a very awkward definition. It is shown here to be equivalent to the following one: truncating $H$ at $n$, the standard stochastic integrals $H_n.X$ converge in the topology of semimartingales. This is clearly invariant under changes of law. A counterexample shows that integrability may be lost if the filtration is enlarged
Comment: See also 1417. This is a synthesis of earlier work, much of which is due to Jacod, Calcul Stochastique et Problèmes de Martingales, Lect. Notes in M. 714. The contents of this paper appeared in book form in Dellacherie-Meyer, Probabilités et Potentiel B, Chap. VIII, \S3. An equivalent definition is given by L. Schwartz in 1530, using the idea of ``formal semimartingales''. For further steps in the same direction, see Stricker 1533
Keywords: Stochastic integrals
Nature: Exposition, Original additions
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