Transformations browniennes et compléments indépendants: résultats et problèmes ouverts

Abstract

In the first part the natural filtration of an R ^d-valued Brownian motion B is compared to that of the BM $B\prime = \int_0^ \cdot {HdB}$, where H is previsible in the filtration of B and valued in O _d (R). We show that there exists a r.v. U, independent of B′ and uniformly distributed on [0, 1] or on some finite set, such that σ(B)=σ(B′) V σ(U), provided either of the following two conditions holds:

• when the transform B ↦B′ is “subordinated” to some subdivision of R ₊;

• —|when this transform commutes with Brownian scaling.

The r.y. U encodes the information lost by the transform B ↦ B′. We show that all kinds of information loss are possible: U may take infinitely many or any finite number of values.

In the second part, we study a related question: suppose given a linear BM X′ in the filtration generated by some planar BM (X, Y). Can one find another linear BM independent from X′ and such that the planar BMs (X′, Y′) and (X, Y) generate the same filtration? We give a necessary condition for X′ to have such an independent Brownian complement, and we study some examples.