Regularity of models associated with Markov jump processes

Abstract

Abstract We consider a jump Markov process X = ( X t ) t ≥ 0 X={\left({X}_{t})}_{t\ge 0} , with values in a state space ( E , ℰ ) \left(E,{\mathcal{ {\mathcal E} }}) . We suppose that the corresponding infinitesimal generator π θ ( x , d y ) , x ∈ E {\pi }_{\theta }\left(x,{\rm{d}}y),x\in E , hence the law P x , θ {{\mathbb{P}}}_{x,\theta } of X X , depends on a parameter θ ∈ Θ \theta \in \Theta . We prove that several models (filtered or not) associated with X X are linked, by their regularity according to a certain scheme. In particular, we show that the regularity of the model ( π θ ( x , d y ) ) θ ∈ Θ {\left({\pi }_{\theta }\left(x,{\rm{d}}y))}_{\theta \in \Theta } is equivalent to the local regularity of ( P x , θ ) θ ∈ Θ {\left({{\mathbb{P}}}_{x,\theta })}_{\theta \in \Theta } .