We discuss certain mixed Cauchy-Dirichlet problems for degenerate parabolic equations in the unbounded interval R^+, which are generalizations of the Black-Scholes equation and of the Cox-Ilgersson-Ross equation. We present existence and regularity results for the (C0) semigroups associated to them in suitable spaces of continuous functions on (0, ∞) highlighting, in particular, analogies and differences between the corresponding generators.