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

The transition kernel of an Double-struck capital Rn-valued diffusion or jump diffusion process {Xt} is known to satisfy the Feller property if {Xt} is the solution of an SDE whose coefficients are Lipschitz continuous. This Lipschitz route to Feller falls short if {Xt} is the solution of an SDE whose coefficients depend on a state-dependent regime-switching process {theta 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 {Xt, theta t} with hybrid jumps, i.e. jumps in {Xt} that occur simultaneously with {theta t} switching.