 Solution

Use the common filtration in which the two driving processes are independent Brownian motions and the solutions are adapted. Let $Z=X+Y$, and define predictable bounded weightswith the ratios defined only on $\{Z>0\}$. The nonnegative solutions have $X=Y=0$ when $Z=0$. SetIndependent Brownian drivers have zero cross-variation. Since $h_1^2+h_2^2=1$ even on the zero set, $[B{]}_t=t$, and the Lévy characterization of Brownian motion makes $B$ a Brownian motion. Moreover $\sqrt{Z}{h}_1=\sqrt{X}$ and $\sqrt{Z}{h}_2=\sqrt{Y}$ everywhere. Adding the two original equations therefore proves additivity of independently driven squared Bessel processes:The unit-vector fill on $\{Z=0\}$ is essential: simply dividing by $\sqrt{Z}$ there would leave the Brownian construction undefined.