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

Jacod, Jean

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

Keywords:

extension of filtration; semimartingale; random variable with values in a Lusin space; canonical decomposition; local martingale

Citations:

Zbl 0547.00034