 Solution

Apply the Itô formula to $Y_t=e^{B_t+t/2}$. The time derivative contributes $Y_{t}dt/2$, and the quadratic-variation correction contributes another $Y_{t}dt/2$. ThusBoth 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 $N_t={\int }_0^t{Y}_{s}d{B}_s$. This is a continuous local martingale withThe strong law for Brownian motion gives $B_t/t\to 0$ almost surely, so $\log Y_t/t\to 1/2$, $Y_t\to \infty$, 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 $d{H}_s=Y_s^{2}ds$, hence $Y_{s}ds=Y_s^{-1}d{H}_s$. Integrating the stochastic equation givesThe infinite clock lifetime verifies that this Brownian motion is defined for every nonnegative clock time without an additional extension.