Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve

Abstract

Let σ > 0 , δ ≥ 1 , b ≥ 0 , 0 < p < 1 . Let λ be a continuous and positive function in H l o c 1 , 2 ( R + ) . Using the technique of moving domains (see Russo and Trutnau (2005) [9]), and classical direct stochastic calculus, we construct for positive initial conditions a pair of continuous and positive semimartingales ( R , R ) with d R t = σ R t d W t + σ 2 4 ( δ − b R t ) d t + ( 2 p − 1 ) d ℓ t 0 ( R − λ 2 ) , and d R t = σ 2 d W t + σ 2 8 ( δ − 1 R t − b R t ) d t + ( 2 p − 1 ) d ℓ t 0 ( R − λ ) + I { δ = 1 } 2 d ℓ t 0 + ( R ) , where the symmetric local times ℓ 0 ( R − λ 2 ) , ℓ 0 ( R − λ ) , of the respective semimartingales R − λ 2 , R − λ are related through the formula 2 R d ℓ 0 ( R − λ ) = d ℓ 0 ( R − λ 2 ) . Well-known special cases are the (squared) Bessel processes (choose σ = 2 , b = 0 , and λ 2 ≡ 0 , or equivalently p = 1 2 ), and the Cox–Ingersoll–Ross process (i.e. R , with λ 2 ≡ 0 , or equivalently p = 1 2 ). The case 0 < δ < 1 can also be handled, but is different. If | p | > 1 , then there is no solution.