# Does the noise term in a SDE need to be Gaussian?

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?

• "generated" in what sense? The Brownian motion is the limiting process for a wide range processes that need not be Gaussian, e.g. take an AR(1) time series, the suitably normalized partial sum processes would converge to Bm. – Michael Dec 3 '14 at 4:19
• Thanks for the response! If I understand correctly, you're saying all types of white noise can be simulated using a Wiener process? – RJ Nowling Dec 3 '14 at 19:01
• Rather I am saying that the Wiener process is the limit process of a wide family of process that need not have i.i.d. normal increments, the key assumption is "short memory". – Michael Dec 4 '14 at 0:19
• Of course, one can talk about Levy-driven, rather than just Wiener-driven, SDE's but that seems to be a different question than the one you're asking. – Michael Dec 4 '14 at 0:20

Let $N_t$ and $M_t$ be independent Poisson processes with equal intensity $\lambda$. Then if we define the Skellam process as $S_t = N_t - M_t$ it has expectation zero and covariance $C[S_{t+h}, S_t] = \lambda t$.