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Zbl 0646.60041
Letta, Giorgio
Un example de processus mesurable adapté non progressif. (An example of a non-progressive adapted measurable process). (French)
[A] Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 449-453 (1988).

[For the entire collection see Zbl 0635.00013.] \par On a probability space endowed with a filtration satisfying the ``usual conditions'' of completeness and right-continuity, if we consider a process $X\sb t=\int\sp{t}\sb{0}H\sb sds$, H measurable and adapted, it is well known that there exists a progressive (even predictable) process whose paths are equivalent to the corresponding paths of H (for the Lebesgue measure $\lambda)$; hence X is adapted. \par In this paper it is first proved that these results are false when we have a space without probability. A counter example is notably proved. Nevertheless a weaker result is obtained in the case of a probability space ($\Omega$,${\cal F},P)$ even without the ``usual conditions''. More precisely, if H is measurable and satisfies $$ \int\sp{t}\sb{0}\vert H(s,\omega)\vert ds<\infty \quad \forall (t,\omega)\quad and\quad X(t,\omega)=\int\sp{t}\sb{0}H(s,\omega)ds, $$ there is equivalence between \par (a) H is equivalent for $\lambda\otimes P$ to a predictable process; \par (b) H is equivalent for $\lambda\otimes P$ to a measurable adapted process; \par (c) X has an adapted modification; \par (d) X is indistinguishable from a predictable process.
[J.P.Lepeltiér]

MSC 2000:: *60G05 Foundations of stochastic processes
60G07 General theory of stochastic processes

Keywords: filtration; counter example; predictable process; adapted modification

Citations: Zbl 0635.00013

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