Is there uniform stochastic process?

Question

Shall I construct a stochastic process $X(t)$ such that $X(s+t)-X(s)\sim U(-t,t)$ ? Or is there already any similar formula?

Answer 1

Let $U$ be uniform in $[-1,1]$ and let $X_t=Ut$, which is uniform in $[-t,t]$. Then $$X_{t+s}-X_s=U(t+s)-Us=Ut$$ so this works. So it's not so much a stochastic process as just a random variable giving the slope of a line.