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When paths move only by jumps of size $\pm 1$, there is neither a Gaussian component nor a continuous drift. The Lévy measure is supported on $\{-1,1\}$ and has finite mass because these points are bounded away from zero. The process is therefore a Compound Poisson process, expressible as a difference of independent Poisson processeswhere the rates are ${\lambda }_{+},{\lambda }_{-}$ respectively. The martingale assumptions implyHence $\boxed{{\lambda }_{+}={\lambda }_{-}=1/2}$. The embedded jump chain is a simple symmetric random walk, and the total jump rate is one. In particular,This is the symmetric Skellam distribution, with the explicit mass functionFor $k\ge 0$, sum the joint probabilities of $N_{+}=j+k,N_{-}=j$; for $k<0$, interchange the two variables. The characteristic function is $\mathbb{E}{e}^{iu{X}_1}=\exp (\cos u-1)$, consistent with the rate-one symmetric Poisson difference process. The condition that paths move only by jumps is essential to exclude a separate continuous component.

Take two independent rate-one Poisson processes $N^{+}$ and $N^{-}$, and letThe difference of independent Poisson processes starts at zero and has stationary increments and independent increments. Its paths are càdlàg. For an interval of length $h$, the probability of any jump is $1-e^{-2h}\to 0$, proving stochastic continuity. Thus $X$ is a Lévy process. The characteristic function of a Poisson distribution with mean $t$ is $\exp (t(e^{iu}-1))$, so independence givesThe sample paths are integer-valued step functions with jumps $+1$ or $-1$. On every bounded interval there are only finitely many jumps, and independent Poisson arrival times coincide with probability zero. The combined arrival rate is two: holding times are independent exponentials of rate two, and each jump direction has probability $1/2$, independently of the holding times. This is equivalently a Compound Poisson process of rate two with Rademacher distribution jump sizes. Its paths have finite variation on compact time intervals, although there are infinitely many jumps over the whole half-line almost surely.