Let $X_t$ be a Poisson process.
Define $Y_t = X_t - t$.
Note that $\Bbb{E}[X_t^k] = \sum_i \left\{{k \atop i}\right\} t^i$, where $\left\{{k \atop i}\right\}$ denote Stirling numbers of the second kind.
Now compute $$\Bbb{E}[Y_t] = \Bbb{E}[X_t - t] = 0 $$ $$\Bbb{E}[Y_t^2] =\Bbb{E}[X_t^2 - 2 X_t t +t^2] = \Bbb{E}[X_t^2] - t^2 = t $$
If we compute a higher moment (please double check) $$\Bbb{E}[Y_t^4] = \Bbb{E}[X_t^4] - 4t \Bbb{E}[X_t^3] + 6 t^2\Bbb{E}[X_t^2] - 4t^3\Bbb{E}[X_t] +t^4 = 3 t^2 +t $$
The last equality comes from the fact that
\begin{align*} \Bbb{E}[X_t^4] &= t^4 + 6 t^3 + 7 t^2 + t\\ -4t\Bbb{E}[X_t^3] &= -4t( t^3 + 3 t^2 + t)\\ 6t^2\Bbb{E}[X_t^2] &= 6t^2( t^2 + t)\\ -4t^3\Bbb{E}[X_t] &= -4t^3( t) \end{align*}
In light of this, for the compensated Poisson process,
$$\Bbb{E}[(Y_t-Y_s)^4] \leq K (t-s)^2$$
Can we apply Kolmogorov's continuity theorem to conclude that $Y_t$ admits a continuous modification?
There is probably an error in my computations, or I am missing something in the hypothesis of Kolmogorov's continuity theorem.
But I don't see it.
Any ideas?