Stochastic Differential Equations (SDEs) appear in many areas of science and engineering. Classical SDEs are differential equations with additional random forcing terms, modelled as white noise. Two of the best known examples are Geometric Brownian Motion, a simple and commonly used model for the evolution of stock prices, and the Langevin equation, a model of a molecular system especially in thermal equilibrium.
Differential equations with random forcing or other random input data that are correlated in space and time (and so, not modelled by white noise) can be handled with standard calculus. Such equations arise in many engineering applications, where uncertain parameters and input data for PDE models (material properties, domain geometries etc.) are modelled as random variables and random fields.
The School of Mathematics at the University of Manchester has a large group working on both types of stochastic differential equations. In the numerical analysis group, research is ongoing into uncertainty quantification and finding fast and accurate methods for solving PDEs with random data using finite element techniques. Research into the underlying probability theory for classical S(P)DEs, especially stochastic analysis and stochastic calculus is being undertaken in the probability group. Finally, there is an active group working on the application of SDEs in mathematical finance.
Recent work on the numerical analysis of SDEs includes
- a priori error estimation for finite element solutions to elliptic PDEs with random diffusion,
- numerical studies of Navier-Stokes fluid flows with random viscosity,
- ergodic theory and the study of long time behaviour of numerical approximations of SDEs,
- analysis of algorithms for stochastic PDEs by Hilbert space theory,
- numerical methods for dissipative particle dynamics, stochastic delay differential equations, and stochastic PDE models of excitable media,
- software development in collaboration with NAG featuring state of the art algorithms for stochastic and random differential equations.
If you are interested in pursuing research in any of these areas, contact Dr. Catherine Powell.