I want to show joint (strong) Markov property of $(Z, R)$ for $(Z_t)$ a Markov chain and $R_t:= \int_0^t Z_s dB_s$ the integral of $Z$ against a standard Brownian motion $B$.

In particular, $Z$ is a time homogeneous Markov chain over continuous time $t\in\mathbb{R}_+$ with discrete state space $\mathcal{S}\subset \mathbb{R}$, càdlàg paths and no explosions (so in particular $Z$ is a semimartingale in the sense of e.g. Protter: Stochastic Integration and Differential Equations).

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What I have so far:

It is pretty straightforward to show (and I guess rather well known) that such a pure jump process $Z$ and a Brownian motion $B$ are independent, the argument bases more or less on the fact that the quadratic covariation $[Z,B]$ vanishes.

Now, I know of various results that link (strong) Markov property to being the strong solution to a stochastic differential equation, but for cases like mine they usually require some stronger assumptions on $Z$ like for example being a Lévy process (e.g. same Protter reference, Chapter V Section 6). Am I correct that a Markov chain $Z$ as defined above need not be a Lévy process?

I am open to other ways of showing joint Markov property, but the approach seems tempting as I already have the generator for $(Z,R)$ given by

$(\mathscr{L}g)(r,z) = \frac12 z^2 \partial^2_{rr}(r,z) + (Qg(r,\cdot))(z)$

for suitable functions $g$ twice differentiable in $r$ and bounded in $z$, and $Q$ the generator (Q-matrix) of the Markov chain $Z$.

Any help is greatly appreciated.


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