Squared Bessel process ($d{Z}_t=2\sqrt{Z_t}d{B}_t+\delta dt$)

A squared Bessel process of dimension $\delta \ge 0$ is a nonnegative continuous solution of the displayed stochastic differential equation. It extends the squared radial part of integer-dimensional Brownian motion to arbitrary nonnegative dimensions. Before hitting zero its square root solves $dR=dB+(\delta -1)(2R{)}^{-1}dt$. At dimension zero, a process started at zero remains there. The coefficients are not Lipschitz at zero, so positivity or uniqueness must not be justified by a global Lipschitz theorem.

For independently driven nonnegative squared Bessel processes $X,Y$, put $Z=X+Y$. Where $Z>0$, use Brownian weights $\sqrt{X/Z}$ and $\sqrt{Y/Z}$. Where $Z=0$, fill the weights with $(1,0)$. Their squared norm is always one and their cross-variation is zero, so their stochastic integral is Brownian motion by the Lévy characterization of Brownian motion. Multiplying by $2\sqrt{Z}$ recovers the sum's martingale part even at zero, and the drifts add to $\alpha +\beta$.