 Solution

The drift $v/2$ is continuously differentiable and hence locally Lipschitz on the open interval $(0,1)$, while the diffusion coefficient is the constant one. The local existence and pathwise uniqueness theorem for a stochastic differential equation therefore gives a unique strong solution up to its first exit from every compact subinterval. These solutions agree by pathwise uniqueness, producing a unique maximal local solution of a stochastic differential equation whose lifetime is