## Grossissement initial, hypothèse (H’) et théorème de Girsanov.(French)Zbl 0568.60049

Grossissements de filtrations: exemples et applications, Sémin. de Calcul stochastique, Paris 1982/83, Lect. Notes Math. 1118, 15-35 (1985).
[For the entire collection see Zbl 0547.00034.]
Let ($$\Omega$$,$${\mathcal F},P)$$ be a complete probability space with filtration $$F=({\mathcal F}_ t)_{t\geq 0}$$ satisfying the usual conditions. Let L be a random variable with values in a Lusin space (E,$${\mathcal E})$$. Consider a new filtration $$G=({\mathcal G}_ t)_{t\geq 0}$$ defined by the relation $${\mathcal G}_ t=\cap_{s>t}[{\mathcal F}_ s\vee \sigma (L)].$$
The author proves that each F-semimartingale is a G-semimartingale if the following condition is satisfied: for each t there exists a positive $$\sigma$$-finite measure $$\eta_ t$$ on (E,$${\mathcal E})$$ such that $$Q_ t(\omega,\cdot)\ll \eta_ t(\cdot)$$ a.s. in $$\omega$$ where $$Q_ t(\omega,dx)$$ is a regular version of the conditional law of L with respect to $${\mathcal F}_ t$$. A canonical decomposition is also given for each F-local martingale.
Reviewer: L.Gal’chuk

### MSC:

 60G44 Martingales with continuous parameter 60G30 Continuity and singularity of induced measures 60G48 Generalizations of martingales

Zbl 0547.00034