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XIII: 36, 427-440, LNM 721 (1979)
YOR, Marc
Les filtrations de certaines martingales du mouvement brownien dans ${\bf R}^n$ (Brownian motion)
The problem is to study the filtration generated by real valued stochastic integrals $Y=\int_0^t(AX_s, dX_s)$, where $X$ is a $n$-dimensional Brownian motion, $A$ is a $n\times n$-matrix, and $(\,,\,)$ is the scalar product. If $A$ is the identity matrix we thus get (squares of) Bessel processes. If $A$ is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of $A$. In particular, this dimension is $1$ if and only if the matrix is equivalent to $cI_r$, a diagonal with $r$ ones and $n-r$ zeros. This is also (even if the symmetry assumption is omitted) the only case where $Y$ has the previsible representation property
Comment: Additional results on the same subject appear in 1545 and in Malric Ann. Inst. H. Poincaré 26 (1990)
Keywords: Stochastic integrals
Nature: Original
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