Parameter Estimation for Rough Differential Equations with Applications to Multiscale Modelling

This proposal seeks support for a PDRA (one year), Dr. Christophe Ladroue. The first goal is to develop and implement methods for statistical estimation and inference in the context of differential equations driven by rough paths (RDEs). Such differential equations are very general and can be used to model a large number of dynamical systems. Our motivation comes from multiscale modelling as we hope the approach we will develop could be useful for important applications, such as molecular dynamics or climate modelling. The second goal is to approximate the slow process of a multiscale stochastic differential equation (SDE) by an RDE at small scales. The final goal would be to use our parameter estimation methods to get some information about the limiting dynamics of the slow process using short paths of the multiscale system.

The principal beneficiaries are the applied mathematics and statistics communities and applied scientists or other practitioners that need to model a stochastic process. We give a detailed description of the ways we hope the applied mathematics and statistics communities to benefit in the case for support. The software package we will produce would be a powerful and versatile tool for modelling random time series (stochastic processes) needed in a large number of fields, varying from economics to telecommunications. This will be publicly available from my website. We will add a demonstration of this software to the The Wolfram Demonstrations Project. We will also present the software to conferences in the applied areas that could benefit from it.