Consider the process $X_t = \sum_{i=1}^{N_t} Y_i$. This is a Lévy process, hence Markov and so on ($N_t$ is a Poisson counting process). Now add some diffusion $D$ for each jump $Y_i$ that starts at the jump time $\tau_i$:
$\tilde{X}_t=\sum_{i=1}^{N_t} Y_i D^i_{t-\tau_i}$
Assume e.g. each $D_i$ to be a geometric Brownian motion. So it is more or less like having a random sum of GBM with random starting points. How can I decide whether this is a semimartingale or not? Intuitively, it is not Markov, so definitely not a Levy process.