Let W be a $\mathbb{R}$-valued Brownian motion. To prove that $(B_t)_{t\geq 0}$, where: $B_t:=W_t-\int_0^t\frac{W_u}{u}du$, is a Brownian Motion with respect to $\mathcal{F}^B$, I showed $[B]_t=t$ and did use Levy's Characterization. Is now $B_t$ also a Brownian Motion with w.r.t. $\mathcal{F}^W$ ? I think showing that $B_t$ is a local $\mathcal{F}^W$-Martingale should be enough. Can i somehow rewrite the equation with Ito's Lemma?
2026-04-06 15:01:02.1775487662
Brownian Motion with Levy's Characterization 2
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