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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.