 Solution

The drift $b(z)=\tanh z$ has derivative $b^{\prime }(z)=1/{\cosh }^{2}z$, so $|b^{\prime }(z)|\le 1$. Hence it is globally Lipschitz. The diffusion coefficient $\sigma (z)=1$ is globally Lipschitz as well, and both coefficients satisfy a linear growth bound.

The global existence theorem for stochastic differential equations with Lipschitz coefficients states that globally Lipschitz coefficients with linear growth give, for each deterministic initial point, an adapted continuous strong solution of a stochastic differential equation on every finite interval, with pathwise uniqueness and no finite-time explosion. Applying this theorem gives a unique strong solution for every $X_0=x\in \mathbb{R}$, satisfying