EPSRC funded project:Inference for Diffusions and Related Processes

Paul Fearnhead and Gareth Roberts

Department of Mathematics and Statistics, Lancaster Univesrity
and
Department of Statistics, University of Warwick.

Summary

Traditional methods for diffusion simulation and related Monte Carlo methods have relied on time-discretisation techniques. This approach has two significant disadvantages: it is usually approximate, and time increments typically need to be small to ensure adequacy of the approximation, and thus methods can be computationally expensive. Recent new methodology for this problem has circumvented the need to disretise time by the use of a powerful and flexible new simulation idea known as Retrospective Sampling. This methodology produces exact simulations (to the accuracy constraints of any computer used for the experiment) and has remarkable efficiency properties, so that there appears to be no cost for exactness in this case. However the Exact Algorithm (EA) framework can be applied only for certain classes of diffusion processes (although this class essentially includes all one-dimensional non-explosive diffusions) This project aims to extend the framework above to a very rich and diverse class of stochastic processes, such as jump diffusions, hypo-elliptic diffusions and solutions of stochastic partial differential equations). The approach is to work both with pure simulation methodology and also with related (and more flexible) importance sampling techniques. There are many potential applications of these methods in scientific problems. We will focus on two important areas. The use of diffusion-related models in Systems Biology is expanding rapidly, and we will apply our methodology here. Secondly, we will consider the problem of rare event simulation in molecular dynamics simulation.

Aims/Objectives

To develop a very general framework for exact simulation of diffusion related stochastic processes, including jump diffusions, diffusions with discontinuous drifts, hypo-elliptic diffusions, and solutions of SPDEs;
To design and investigate (both theoretically and empirically) Monte Carlo methodology for the unbiased estimation of arbitrary expected diffusion functionals and related stochastic integrals;
To investigate links between the methodology developed in 1 and 2 and other importance sampling methodologies;
To consider algorithms for joint updating of parameters and missing data in diffusion filtering problems;
To consider applications of the above methodology, notably in Systems Biology and rare event simulation;
To organise a workshop in the area of the project for the benefit of the world-leading UK community in modelling and inference for stochastic processes.

Publications

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