= Solution
Let $Y_t=-\beta\int_0^tB_u\,dB_u-\int_0^th_u\,du$. Its <quadratic variation> is $[Y]_t=\beta^2\int_0^tB_u^2du$. The <Itô formula> for $Z=e^Y$ gives
$$
dZ_t=-\beta Z_tB_t\,dB_t
+Z_t\left(\frac{\beta^2}{2}B_t^2-h_t\right)dt.
$$
Choose
$$
\boxed{h_t=\frac{\beta^2}{2}B_t^2,\qquad dZ_t=-\beta B_tZ_t\,dB_t,\quad Z_0=1.}
$$
The chosen process is continuous and <adapted>, hence <progressively measurable>. Its time integral is finite on compact intervals because Brownian paths are continuous. The resulting $Z$ is the <stochastic exponential> of $-\beta\int_0^\cdot B_u\,dB_u$, and is a positive <local martingale>.
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