 Solution

Let $Y_t=-\beta {\int }_0^t{B}_{u}d{B}_u-{\int }_0^t{h}_{u}du$. Its quadratic variation is $[Y{]}_t={\beta }^2{\int }_0^t{B}_u^{2}du$. The Itô formula for $Z=e^Y$ givesChooseThe 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 $-\beta {\int }_0^{\cdot }{B}_{u}d{B}_u$, and is a positive local martingale.