Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve

Let σ,δ>0,b≥0σ,δ>0,b≥0. Let λ2:R+→R+λ2:R+→R+, be continuous, and locally of bounded variation. We develop a general analytic criterion for the pathwise uniqueness of Rt=R0+∫0tσ|Rs|dWs+∫0tσ24(δ−bRs)ds+(2p−1)ℓt0(R−λ2), where p∈(0,1)p∈(0,1), and ℓt0(R−λ2) is the symmetric semimartingale local time of R−λ2R−λ2. The criterion is related to the existence of nice (Kummer) functions for the time dependent infinitesimal generator of RR. As a corollary we obtain explicit sufficient conditions for pathwise uniqueness. These are expressed in terms of λ2λ2, its derivative, and the parameters σ,δ,b,pσ,δ,b,p.