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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.

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Why isn't it Markov? Reminds me of a construction of diffusion with jumps which is a Markov process. –  Ilya Aug 3 '12 at 12:04
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Take $D$ as GBM as above. The variance of a GBM depends nonlinear on its starting point (here $Y_i$), cf. en.wikipedia.org/wiki/Geometric_Brownian_motion. So it makes a difference if you have two jumps of, say $y_1$ and $y_2$, or just one jump of $y_1+y_2$. So assume you have only one jump and at time $t$: $L_t = l$. Then for $L'_t$ which had two jumps but also equals $l$, the Markov condition should be violated. It does matter if you depend on $L_s$ or $\mathcal{F}_s$. –  user13655 Aug 3 '12 at 12:46

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