= Solution
The drift $b(z)=\tanh z$ has derivative $b'(z)=1/\cosh^2z$, so $|b'(z)|\le1$. 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 \b[a unique strong solution for every $X_0=x\in\mathbb R$], satisfying
$$
X_t=x+\int_0^t\tanh X_s\,ds+W_t.
$$
Back to article page
