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Let $X_t$ be an Itō process related to a Brownian motion $B_t$

$$ X_t = \int_0^t a(t) \text{d}B_t $$

where $a(t)$ is a process which is adapted to the natural filtration of $B_t$, say $\mathcal{B}_t$.

Considering the natural filtration $\mathcal{X}_t$ of the process $X_t$, what simple (simple-to-prove) conditions warrant that the filtrations $\mathcal{X}_t$ and $\mathcal{B}_t$ are identical? For instance if $a(t)$ is a non-random positive function, this seems to be true.

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  • $\begingroup$ Maybe if you can write $B_t=\int_0^t b(s)dX_s$ for a suitable process $b$ $\mathcal{X}_t$-adapted ? $\endgroup$
    – TheBridge
    May 30, 2019 at 9:14
  • $\begingroup$ $b(s):= 1/a(s)$ seems the only choice. But is-it $\mathcal{X}_t$-adpated? $\endgroup$
    – Yves
    May 30, 2019 at 11:05
  • $\begingroup$ I think you know what to look for now $\endgroup$
    – TheBridge
    May 30, 2019 at 19:20
  • $\begingroup$ may be you should start with simple processes $\endgroup$
    – TheBridge
    May 30, 2019 at 19:21

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