Stochastic ordinary differential equations in applied and computational mathematics

Using concrete examples, we discuss the current and potential use of stochastic ordinary differential equations (SDEs) from the perspective of applied and computational mathematics. Assuming only a minimal background knowledge in probability and stochastic processes, we focus on aspects that distinguish SDEs from their deterministic counterparts. To illustrate a multiscale modelling framework, we explain how SDEs arise naturally as diffusion limits in the type of discrete-valued stochastic models used in chemical kinetics, population dynamics and, most topically, systems biology. We outline some key issues in existence, uniqueness and stability that arise when SDEs are used as physical models and point out possible pitfalls. We also discuss the use of numerical methods to simulate trajectories of an SDE and explain how both weak and strong convergence properties are relevant for highly efficient multilevel Monte Carlo simulations. We flag up what we believe to be key topics for future research, focussing especially on non-linear models, parameter estimation, uncertainty quantification, model comparison and multiscale simulation.