Regularity of models associated with Markov jump processes - Jedidi Wissem (Open Mathematics, 2022)
We consider a jump Markov process X=(Xt)t≥0X={\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 πθ(x,dy),x∈E{π }_{θ }\left(x,{\rm{d}}y),x\in E, hence the law Px,θ{{\mathbb{P}}}_{x,θ } of XX, depends on a parameter θ∈Θθ \in Θ . We prove that several models (filtered or not) associated with XX are linked, by their regularity according to a certain scheme. In particular, we show that the regularity of the model (πθ(x,dy))θ∈Θ{\left({π }_{θ }\left(x,{\rm{d}}y))}_{θ \in Θ } is equivalent to the local regularity of (Px,θ)θ∈Θ{\left({{\mathbb{P}}}_{x,θ })}_{θ \in Θ }.