I'm working on a PhD project that involves parameter estimation for diffusion processes. I'm based in a machine learning research group, and the emphasis here is strongly on "practical" research.

I've developed some theory, and now I'm starting to look for real-world problems to apply it to. To this end, I'd like to ask for some examples of phenomena that are 'naturally' modelled as diffusion processes. An ideal answer would include some justification of why the continuous-time setting is more appropriate than, say, a discrete-time Markov chain.

Two great examples of the kind of thing I'm looking for can be found at the Azimuth project website (here and here). The first article discusses a noisy analogue of a dynamical system that exhibits a Hopf bifurcation. It's suggested that this system might be a sensible first step in modelling oscillatory weather patterns such as El Nino. The second article is somewhat related, and discusses noisy predator-prey systems.

Thanks in advance for your help.