# Diffusion processes in probabilistic modelling

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.

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Please also have a look at the page about stochastic resonance on Azimuth (there are links to review papers in the reference section of that page).

While stochastic resonance has been "invented" to explain the glacial cycles, there are a lot of other systems that exhibit this phenomenon. As an exercise in parametric estimation, you could try to estimate the model parameters of a bistable symmetric potential to temperature series of Earth's history.

It is also quite customary to model stochastic resonance both in continuous time and space and as a discrete system (with discrete time or as a discrete two state Markov system in continuous or discrete time). So this is a natural playing ground for comparing both approaches.

In the case of the glacial cycles there are several periodic or quasi periodic external forcings like the Milankovich cycles, which are most naturally modelled in continuous time. In simulations you'll always use a discrete approximation, of course, so a continuous time model should in this context be viewed as a means to change the time step in your approximation according to your needs, which is a kind of modelling freedom that a discrete Markov model does not have. The continuous time model allows you to compare results obtained for different discretizations, while you have to build in the discretization into a discrete Markov model a priori without any chance to check if that is a good approximation.

Edit, Addendum: Two books written for practitioners in physics and other natural sciences with lots of applications of diffusion processes are

• Hannes Risken: The Fokker-Planck equation. Methods of solution and applications.

• Crispin Gardiner: Stochastic methods. A handbook for the natural and social sciences.

I think both are also cited on the Azimuth project.

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Interesting, thanks for the response. – Simon Lyons Aug 23 '11 at 11:50
Another good reference is van Kampen's Stochastic Processes in Physics and Chemistry. – Paul Tupper Aug 24 '11 at 16:00

You might look at the introduction of Bernt Oskendal's Stochastic Differential Equations. He gives seven motivating problems (at least he does in the edition I have) for studying stochastic calculus. One big area that you have not mentioned is the applications of stochastic calculus to finance.

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My personal feeling is that financial time series aren't modelled particularly well by diffusion processes. Increments tend to exhibit high kurtosis, which means they're better described with a Levy-driven SDE rather than a process with locally Gaussian increments. The Markov property can potentially cause problems, too. – Simon Lyons Aug 22 '11 at 18:42
Well even if increments exhibit high moments, you can still extend the diffusion framework to so-called stochastic volatility, which is an active area of estimation under the additional constraint of Hidden Markov process. Regards – The Bridge Aug 22 '11 at 19:46