One very common and simple model is to write that the returns $$\frac{ X(t) - X(t-\delta t)}{X(t-\delta t)} $$ are gaussian with some standard variantion $\sigma (\delta t)$. If you live in a no-free lunch world (meaning that there is way to make money without taking risks: this is convenient except for high frequency time series), then the returns of non-overlapping periods are uncorrelated.
Finally, note that empirical studies reveal that $\sigma (\delta t) \propto \sqrt{\delta t}$. This leads to a geometric Brownian motion model.