= Solution
When paths move only by jumps of size $\pm1$, 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 processes>
$$
X_t=N_t^+-N_t^-,
$$
where the rates are $\lambda_+,\lambda_-$ respectively. The martingale assumptions imply
$$
0=\mathbb E X_t=(\lambda_+-\lambda_-)t,\qquad
\operatorname{Var}(X_t)=(\lambda_++\lambda_-)t=t.
$$
Hence $\boxed{\lambda_+=\lambda_-=1/2}$. The embedded jump chain is a <simple symmetric random walk>, and the total jump rate is one. In particular,
$$
\boxed{X_1\overset d=N_+-N_-,\qquad N_+,N_-\text{ independent Poisson}(1/2).}
$$
This is the symmetric <Skellam distribution>, with the explicit mass function
$$
\boxed{\mathbb P(X_1=k)=e^{-1}\sum_{j=0}^{\infty}\frac{2^{-(2j+|k|)}}{j!(j+|k|)!},\qquad k\in\mathbb Z.}
$$
For $k\geq0$, sum the joint probabilities of $N_+=j+k,N_-=j$; for $k<0$, interchange the two variables. The <characteristic function> is $\mathbb E e^{iuX_1}=\exp(\cos u-1)$, 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.
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