| Popis: |
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. |
