= Solution
The drift $v/2$ is continuously differentiable and hence <locally Lipschitz function>[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
$$
\mathcal T=\inf\{t\geq0:X_t\in\{0,1\}\}.
$$
Back to article page
