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
$$
\mathcal T=\inf\{t\geq0:X_t\in\{0,1\}\}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 5 / a / Solution

A function is locally Lipschitz when every point has a neighborhood on which the function satisfies a <Lipschitz bound>. Continuously differentiable functions are locally Lipschitz.

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