# Itō process and filtrations

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.

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