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

Let σ , δ > 0 , b ≥ 0 . Let λ 2 : R + → R + , be continuous, and locally of bounded variation. We develop a general analytic criterion for the pathwise uniqueness of R t = R 0 + ∫ 0 t σ | R s | d W s + ∫ 0 t σ 2 4 ( δ − b R s ) d s + ( 2 p − 1 ) ℓ t 0 ( R − λ 2 ) , where p ∈ ( 0 , 1 ) , and ℓ t 0 ( R − λ 2 ) is the symmetric semimartingale local time of R − λ 2 . The criterion is related to the existence of nice (Kummer) functions for the time dependent infinitesimal generator of R . As a corollary we obtain explicit sufficient conditions for pathwise uniqueness. These are expressed in terms of λ 2 , its derivative, and the parameters σ , δ , b , p .