Most of the examples I've seen for stochastic differential equations are of the form:
$$ dX_t = \mu(X_t, t)dt + \sigma(X_t, t) dW_t $$
where $dW_t$ is a Wiener process, i.e., the independent increments are normally distributed.
I played around a little bit with different distributions, looking at the results using autocorrelation functions and the FFT. It seems that white noise could also be generated by sampling from a uniform distribution, whereas sampling from an exponential distribution produced colored noise due to the monotonically increasing values of $X_t$.
So, what are the requirements for the distribution for sampling the independent intervals in order to produce white noise? (e.g., needs to span $[-\infty, \infty]$, sum of probabilities for values < 0 needs to equal sum for values > 0, etc.)
Are there any good examples that do not use Gaussian distributions?