Feller property of regime-switching jump diffusion processes with hybrid jumps

The transition kernel of an ℝ ⁿ-valued diffusion or jump diffusion process {X _t} is known to satisfy the Feller property if {X _t} is the solution of an SDE whose coefficients are Lipschitz continuous. This Lipschitz route to Feller falls short if {X _t} is the solution of an SDE whose coefficients depend on a state-dependent regime-switching process {θ _t}. In this paper it is shown that pathwise uniqueness and the Feller property are satisfied under mild conditions for a regime-switching jump diffusion process {X _t, θ _t} with hybrid jumps, i.e. jumps in {X _t} that occur simultaneously with {θ _t} switching.