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

Abstract

The transition kernel of an ℝⁿ-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.

Keywords:

• Feller property
• hybrid jumps
• hybrid state Markov process
• Itô-Skorohod stochastic differential equation
• strong solution

AMS Subject classification (2010)::

• 60G57
• 60H20
• 60J05

Acknowledgement

The author would like to thank anonymous reviewers for comments and suggestions that were very helpful in improving the paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.