# A Lévy process is a semimartingale proof

I have to prove that a Lévy process is a semimartingale.

In general we say that $$X$$ is a semimartingale if it is an adapted process such that, for each $$t ≥ 0$$,

$$X (t) = X (0) + M(t) + C(t)$$ where $$M = (M(t), t ≥ 0)$$ is a local martingale and $$C = (C(t), t ≥ 0)$$ is an adapted process of finite variation.

By the Lévy–Itô decomposition we have, for each $$t ≥ 0$$,

$$X (t) = M(t) + C(t)$$,

where $$M(t) = B_A(t) + \int_{|x|\leq1} x \tilde{N} (t, dx)$$ and $$C(t) = bt +\int_{|x|>1} xN(t, dx)$$, with $$\int_{A} xN(t,dx)= \sum_{0 \leq u \leq t} \Delta X_u 1_A(\Delta X_u)$$ and $$\int_A x\tilde{N}(t,dx):= \int_A xN(t,dx)-t\int_A x\nu(dx).$$

Now, I think I can prove that $$M$$ is a martingale and that $$C$$ is of finite variation, but with respect to which filtration?

Because the book "Lévy Processes and Stochastic Calculus" of Applebaum doesn't say anything about the filtration but in another book I read that $$M$$ and $$C$$ are to be adapted to the same filtration to which X is adapted.

But, for example I can prove that $$(B_A(t),t\geq 0)$$ is a martingale wrt his natural filtration, not wrt the filtration to which the Lévy processes $$X$$ is adapted.

Similarly, $$\int_{|x|\leq1} x \tilde{N} (t, dx)$$ is a martingale wrt his natural filtration?

Thanks.

The statement that cadlag Levy is semimartingale is proved here in theorem 4. Namely they prove that a cadlag Lévy process $$X$$ decomposes as $$X_t=bt+W+Y$$ where $$Y$$ is a semimartingale and $$W$$ is a continuous centered Gaussian process with independent increments, hence a martingale.

Filtration question

When one starts with a semimartingale $$X$$ with filtration $$F_{t}$$, then the Doobs-Meyer decomposition theorem gives decomposition $$X=M+A$$, where these are adapted to $$F_{t}$$ since we use projections of $$X$$ in order to construct them.

Conversely, if one starts with the decomposition, one can use Stricker's theorem

if you take a semimartingale X according to some filtration F and if G is a subfiltration of F for which X is adapted, then X is also a G-semimartingale

Decomposition

Just as an aside, for a semimartingales to have the decomposition you mentioned one needs some additional assumptions eg. see MSE answer here

From Kal97, pg. 446:

Theorem 23.14 Any semimartingale $$X$$ has an a.s. unique decomposition $$X=X_0 + X^c + X^d$$ where $$X^c$$ is a continuous local martingale with $$X_0^c=0$$ and $$X^d$$ is a purely discontinuous semimartingale.

And if we want to have $$X^{d}$$ further be finite variation one needs

$$\sum_{s\le t}\vert \Delta X_s\vert < \infty\text{ (a.s.) }.$$

(eg. Cauchy process is a semimartingale that fails to have it here).

Filtration shrinkage

Also, as another aside there can be issues of losing semimartingale if we project into a smaller filtration. For some discussion and references see here Semimartingale decomposition and filtrations and also online article "Local Martingales and Filtration Shrinkage".