= Solution
Apply the <Itô formula> to $Y_t=e^{B_t+t/2}$. The time derivative contributes $Y_tdt/2$, and the quadratic-variation correction contributes another $Y_tdt/2$. Thus
$$
\boxed{dY_t=Y_t\,dB_t+Y_t\,dt,\qquad Y_0=1.}
$$
Both the drift $b(y)=y$ and diffusion coefficient $\sigma(y)=y$ are globally Lipschitz. The <global existence theorem for stochastic differential equations with Lipschitz coefficients> gives a unique <strong stochastic solution> for a fixed driving <Brownian motion> and initial value; in particular <pathwise uniqueness> holds. The explicit solution is strictly positive.
Write $N_t=\int_0^tY_s\,dB_s$. This is a <continuous local martingale> with
$$
[N]_t=H_t=\int_0^tY_s^2ds=\int_0^te^{2B_s+s}ds.
$$
The <strong law for Brownian motion> gives $B_t/t\to0$ almost surely, so $\log Y_t/t\to1/2$, $Y_t\to\infty$, and $H_t\to\infty$. The <Dambis-Dubins-Schwarz theorem> therefore supplies a <Brownian motion> $\beta$ in the inverse-clock filtration with $N_t=\beta_{H_t}$ for all $t$. Finally $dH_s=Y_s^2ds$, hence $Y_sds=Y_s^{-1}dH_s$. Integrating the stochastic equation gives
$$
\boxed{Y_t=1+\beta_{H_t}+\int_0^t\frac1{Y_s}\,dH_s.}
$$
The infinite clock lifetime verifies that this <Brownian motion> is defined for every nonnegative clock time without an additional extension.
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