Rough differential equations (RDEs) are very general and can be used to model a large number of dynamical systems. Their main difference from models currently used are that they allow for long-range dependence that is often seen in many types of real data coming from a range of systems, such as financial systems, molecular dynamics or the climate. In order to be able to fit these models to data, one needs a statistical inference theory for the estimation of unknown parameters. Once this is done, the behaviour of these models could be studied through the analysis of the fitted RDE which is, in most cases, simpler and easier to analyse than models currently used.
We have developed a moment-matching type of estimator for estimating parameters in the context of differential equations driven by rough paths (RDEs). See Parameter Estimation for Rough Differential Equations (A. Papavasiliou and C. Ladroue). Annals of Statistics 39(4): 2047–-2073, 2011.
The theory is accompanied by a software package that implements the methodology. See A Distributed Procedure for Computing Stochastic Expansions with Mathematica. (C. Ladroue and A. Papavasiliou). J. Stat. Softw. (accepted) for the description of how the code works.
Our ultimate goal is to be able to analyse and predict the behaviour of complex systems through RDEs. To succeed, we plan to replace the multiscale (possibly atomistic) models describing the complex systems by an RDE modelling the long range behaviour of the system. We have studied this problem in the simple context of a multiscale Ornstein-Uhlenbeck process and showed that this is possible. See Coarse-grained modeling of multiscale diffusions: the p-variation estimates. (A. Papavasiliou) Stochastic Analysis 2010, pp 69--90, Springer, 2010.
This research has been funded by EPSRC grant EP/H019588/1.