Multiscale methods for stochastic differential equations and applications

Abstract

This thesis is on multiscale methods for Stochastic Differential Equations (SDEs). We provide theoretical results for the method of averaging for SDEs, as well as an application to parameter inference. We first study Poisson equations and averaging for SDEs, the former as a means of producing results on the latter. We treat Poisson equations on non-compact state spaces for coefficients that can grow super-linearly. This is one of the two building blocks towards the second (and main) result of the thesis, namely we obtain a uniform in time (UiT) averaging result (with a rate) for SDE models with super-linearly growing coefficients. Key to obtaining both our UiT averaging result and to enable dealing with the super-linear growth of the coefficients (both of the slow-fast system and of the associated Poisson equation) is exponential decay in time of the space-derivatives of appropriate Markov semigroups. Motivated by applications to mathematical biology, we then study the averaging problem for slow-fast systems, in the case in which the fast dynamics is a stochastic process with multiple invariant measures. We work in the setting in which the slow process evolves according to an Ordinary Differential Equation (ODE) and the fast process is a continuous time Markov Process with finite state space and show that, in this setting, the limiting (averaged) dynamics can be described as a random ODE (that is, an ODE with random coefficients.) We also present an application of the results presented in this thesis to statistical modelling, namely parameter inference. The content of this thesis has resulted in two papers, one submitted and one published (see [1] and [2]), with a third close to submission.