= Solution
A <strong solution of a stochastic differential equation> is constructed for the prescribed driving <Brownian motion> on the prescribed stochastic basis, adapted to the augmented filtration generated by that driver and the initial data, and satisfies the integral equation almost surely. For this equation it must be continuous, nonnegative and satisfy $Z_t=z+\int_0^t\sqrt{Z_s}\,dB_s$. <Pathwise uniqueness> means that any two solutions on the same filtered space, with the same initial value and the same driving <Brownian motion>, are indistinguishable.
The standard <global existence theorem for stochastic differential equations with Lipschitz coefficients> cannot be applied to the <square-root branching diffusion>, because $\sigma(x)=\sqrt x$ is not Lipschitz at zero:
$$
\frac{|\sigma(h)-\sigma(0)|}{|h-0|}=\frac1{\sqrt h}\longrightarrow\infty\qquad(h\downarrow0).
$$
It is locally Lipschitz on $(0,\infty)$, which gives a solution only up to its possible approach to zero by that theorem. Thus \b[the missing hypothesis is the Lipschitz bound at the absorbing boundary], not any failure of continuity. Failure of this sufficient criterion does not establish failure of strong existence or uniqueness; part (c) constructs strong existence directly.
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