Let $(B,C,\nu)$ be the characteristics of a semimartingale $\{X_t\}_t$ on $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$.

If \begin{equation} B_t(\omega)=bt, \ C_t(\omega)=ct, \ \nu(\omega,dt,dx)=dt \times F(dx), \end{equation} for suitable $b,c,F$, we know that $\{X_t\}_t$ is a Lévy process.

On the other hand, by each $(B,C,\nu)$ with the above representation a Lévy process $\{Z_t\}_t$ is defined.

Now, similar to the statement for Lévy processes, I would like to construct a semimartingale from a given triplet $(B,C,\nu)$ (which is not deterministic).

To be more specific, suppose \begin{equation} \tilde{B}_t=\int_0^t b_s ds, \ \tilde{C}_t=\int_0^t c_s ds, \ \tilde{\nu}(dt,dx)=dtF_t(dx), \end{equation} for $\{b_t\}_t$ a predictable processes, $\{c_t\}_t$ a positive predictable processes, and $F_t=F_t(\omega,dx)$ for each $(\omega,t)$ a measure on $\mathbb{R}$.

Then, motivated by the Lévy–Itô decomposition for Lévy processes, I thought about a process $\{X_t\}_t$ defined by \begin{equation} X_t=\int_0^t b_s ds+\int_0^t \sqrt{c_s}dW_s+\int_0^t\int_{|x|\leq 1}x(\tilde{\mu}-\tilde{\nu})(ds,dx)+\int_0^t\int_{|x|>1}x\tilde{\mu}(ds,dx), \end{equation} where $\tilde{\mu}$ is a random measure with compensator $\tilde{\nu}$.

My question is: Does this construction define a semimartingale with characteristics $(\tilde{B},\tilde{C},\tilde{\nu})$?

In particular, I am not sure if the last integral is a process of finite variation.


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