Summary: Let \(\sigma>0\), \(\delta\geq 1\), \(b\geq 0\), \(0<p<1\). Let \(\lambda\) be a continuous and positive function in \(H^{1,2}_{\text{loc}}(\mathbb R^+)\). Using the technique of moving domains [see F. Russo and G. Trutnau, J. Funct. Anal. 221, No. 1, 37–82 (2005; Zbl 1071.60069)], and classical direct stochastic calculus, we construct for positive initial conditions a pair of continuous and positive semimartingales \((R,\sqrt R)\) with
\[ dR_t=\sigma\sqrt{R_1}dW_t+\frac{\sigma^2}{4}\;(\delta-bR_t)dt+(2p-1)d\ell^0_t(R-\lambda^2), \]
and
\[ dR_t=\frac\sigma 2\;dW_t+\frac{\sigma^2}{8}\left(\frac{\delta-1}{\sqrt{R_t}}-b\sqrt{R_t}\right)dt+(2p-1)d\ell^0_t(\sqrt R-\lambda)+\frac{\mathbb I_{\{\delta=1\}}}{2}\;d\ell^{0+}_t(\sqrt R), \]
where the symmetric local times \(\ell^0(R-\lambda^2)\), \(\ell^0(\sqrt R-\lambda)\), of the respective semimartingales \(R-\lambda^2\), \(\sqrt R-\lambda\) are related through the formula
\[ 2\sqrt R\,d\ell^0(\sqrt R-\lambda)=d\ell^0(R-\lambda^2). \]
Well-known special cases are the (squared) Bessel processes (choose \(\sigma=2\), \(b=0\), and \(\lambda^2\equiv 0\), or equivalently \(p=\frac12\)), and the Cox-Ingersoll-Ross process (i.e. \(R\), with \(\lambda^2\equiv 0\), or equivalently \(p=\frac12\)). The case \(0<\delta<1\) can also be handled, but is different. If \(|p|>1\), then there is no solution.