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Projection of a diffusion on a slow filtration. (Projection d’une diffusion sur sa filtration lente.) (French) Zbl 0857.60077
Azéma, J. (ed.) et al., Séminaire de probabilités XXX. Berlin: Springer. Lect. Notes Math. 1626, 228-242 (1996); Errata Lect. Notes Math 1655, 329 (1997).
Let $$(X_t)$$ be a real diffusion on natural scale. For all $$s\geq 0$$, let $$G_s$$ be the last instant before $$s$$ when $$(X_t)$$ is equal to zero. The slow filtration is defined as a filtration that, at every time $$s$$, knows everything about $$(X_t)$$ until $$G_s$$. We exhibit the expression of the optional projection of $$(X_t)$$ on its slow filtration, involving the Lévy measure of its positive and negative excursions. Then we calculate explicitly several examples. The most famous of them is the Azéma martingale, where the projected process is the linear Brownian motion.
For the entire collection see [Zbl 0840.00041].
Reviewer: C.Rainer (Lannion)

##### MSC:
 60J60 Diffusion processes
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