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The Lévy-Itô decomposition builds a real Lévy process from drift, continuous Gaussian noise and Poisson jumps. For the triplet $(a,b,K)$ in the Lévy-Khintchine formula with small-jump cutoff one, take independent standard Brownian motion $B$ and Poisson random measure $N$ of intensity $dsK(dx)$. ThenHere $b\ge 0$, $K$ is a Lévy measure and $\tilde{N}=N-dsK(dx)$ denotes compensation. The large-jump term has finitely many jumps on each bounded time interval because $K(|x|>1)<\infty$.

The small-jump integral is defined by limits, rather than by subtracting two potentially infinite sums. For $\epsilon >0$, let $J_t^{\epsilon }={\int }_0^t{\int }_{\epsilon <|x|\le 1}x\tilde{N}(ds,dx)$. These are square-integrable martingales. The compensated Poisson isometry givesas $0<\delta <\epsilon \downarrow 0$. Doob's L2 maximal inequality also controls the supremum on $[0,T]$, so the limit exists locally uniformly in probability, indeed in the L2 norm of that supremum. This uses exactly the small-jump second-moment condition of a Lévy measure.^[ref]

The components have independent increments; multiplying their characteristic functions recovers the Lévy-Khintchine exponent. The decomposition separates continuous fluctuations from large jumps and possibly infinitely many small jumps. A finite small-jump Second moment does not require a finite first absolute jump moment, which explains why compensation and its limiting construction are essential.

For $t>0$, use the continuous logarithm of the characteristic function, normalized to zero at $u=0$, to writeThus the deterministic drift is $+c$. In particular the last contribution is not the negative drift of a compensated unit-jump process.

Take a Poisson random measure $N$ on $(0,\infty )\times \{+,-,p\}$ with intensityDefine the jump marks $j(+)=b$, $j(-)=-b$, and $j(p)=1$. A realization with the required process law is the Poisson stochastic integral with finite intensitywhere the three counts are independent Poisson processes of rates $a/2,a/2,c$. Their characteristic functions multiply towhich is exactly the given expression. The constructed process is a Lévy process, and stationary independent increments make its entire finite-dimensional law determined by these one-time characteristic functions. This gives a representation in law of the specified process.

Equivalently, using its nonzero-jump measure, the Lévy measure isand the unmarked representation is $X_t=ct+{\int }_{(0,t]\times (\mathbb{R}\setminus \{0\})}zN(ds,dz)$ with intensity $ds\nu (dz)$. The finite-jump case of the Lévy–Itô decomposition realizes this pathwise using the jump measure of a version of $X$; the integral is an uncompensated finite sum.

The atomic compound Poisson process with drift has càdlàg paths with finitely many nonzero jumps on every bounded time interval, linear slope $c$ between jumps, and no Brownian component. When $b>0$, jumps of sizes $b,-b,1$ occur at the stated rates; when $b=1$, positive unit-jump rates combine to $a/2+c$. When $b=0$, the two symmetric marks have zero effect and are omitted from the Lévy measure; the process reduces to $ct+N_t^p$, so $a$ has no effect. If $c=0$ the paths are piecewise constant, and if also $ab=0$ they are identically zero. All paths have finite variation on bounded intervals. As a check on the drift sign,

Let $B$ be Brownian motion and let $N(ds,dx)$ be an independent Poisson random measure with intensity $dsK(dx)$. Writing $\tilde{N}=N-dsK(dx)$ for its compensated version, the Lévy–Itô decomposition constructsThe four terms are independent drift, Gaussian, compensated small-jump and compound-Poisson large-jump components.

For the small-jump cutoff convention in the Lévy-Khintchine formula, let $(a,b,K)$ be the triplet of a real Lévy process. Its paths are continuous almost surely exactly when $K=0$. They are everywhere differentiable almost surely exactly when $K=0$ and $b=0$, giving the deterministic path $X_t=at$. A compound Poisson path can have a derivative between its jumps without being an everywhere differentiable path.

finite variation on compact time intervals has the different criterionUnder these conditions the uncompensated Lévy-Khintchine formula gives a drift plus an absolutely summable small-jump process and finitely many large jumps. A nonzero Brownian component precludes finite variation, while an infinite absolute small-jump sum already forces infinite variation.

Integrability requires only the large-jump first moment:The small compensated jumps are square integrable and hence integrable. When this condition holds, $\mathbb{E}{X}_t=t[a+{\int }_{|x|>1}xK(dx)]$; the small-jump compensation convention is why no small-jump mean appears.

Likewise a finite second moment is equivalent to $\int x^{2}K(dx)<\infty$, and thenThe Lévy-Itô decomposition proves these moment statements by independent Brownian, compensated-small-jump and compound-Poisson-large-jump parts. The large-jump one-event case also shows why the tail moment conditions are necessary.