The simplest example of the sort of thing you are trying to do would be a stock price that follows a geometric Brownian motion and is subject to jumps at Poisson arrival times. If the jumps are multiplicative, lognormal and have mean equal to 1, then you have three parameters you need to estimate (forget about the drift, at least for now): the regular volatility, the jump volatility, and the intensity of the Poisson process. As a practical matter, I think you need to pick a cut-off, and assume that daily changes smaller than that are diffusive and changes larger than that are the effect of jumps. For a given cut-off, it would seem to be straightforward to estimate the three parameters in question, as well as to do basic specification tests. You should also be able to find the cut-off level that minimizes the probability of misclassifying moves, given the estimated parameters. Regardless of exactly how you go about doing this I would strongly suggest that you explore the properties of your statistical tests on simulated data before starting to draw conclusions about the real world.