= Solution
The relation between $B$ and $W$ is the linear equation $dB_s=dW_s-\beta B_sds$ with $B_0=0$. Multiplication by the deterministic integrating factor $e^{\beta s}$ and the <Itô product rule> give
$$
d(e^{\beta s}B_s)=e^{\beta s}\,dW_s.
$$
Hence
$$
\boxed{B_s=\int_0^se^{-\beta(s-u)}\,dW_u
=W_s-\beta\int_0^se^{-\beta(s-u)}W_u\,du.}
$$
The second equality follows by <semimartingale integration by parts>. It expresses $B$ as an explicit function of the continuous path of $W$, using only an ordinary time integral. This inversion holds before changing measure; $W$ is already a <continuous semimartingale> under the original measure.
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