DK14: Stable and efficient numerical methods for Stochastic Differential Equations

The need to model with and thus to treat SDEs numerically has emerged in many different application areas, such as computational finance, chemical kinetics, laser dynamics, neuroscience, molecular dynamics or electrical circuits. An important topic in deterministic numerical analysis is concerned with how well the numerical method, used with a finite non-zero step-size, reproduces the qualitative behaviour of the analytic solutions. There exists a considerable body of theory and practice for deterministic systems and methods, such as linear and nonlinear stability analysis or geometric integration methods. For the case of SDEs, this type of investigations is in its early stages, but a number of contributions devoted to linear stability analysis of numerical methods exists, in particular for systems of SODEs.  The subject of numerical methods for SPDEs is rather more recent and, although various numerical methods have been proposed and analytically studied for their convergence, only very little has been done in the area of practical implementations of these methods and the related questions of efficiency and stability of the algorithms.

 

The importance of efficient algorithms becomes especially evident when expectations of functionals of solutions of SPDEs have to be computed, i.e, using Monte Carlo methods, which are well-known to have slow convergence properties and high computational complexity. Multi-level Monte Carlo methods are a state-of-the-art approach to reduce the complexity of the original Monte Carlo methods, where again the question of stability of the numerical methods arises when choosing methods for the time-discretisation and their discretisation parameters. The subject of this project is the investigation of essential issues concerning the relationship between stability and convergence of numerical methods for SPDEs in the context of implementations of the Multi-level Monte-Carlo method. A highly relevant issue in this context is the characterisation of stiffness in SDEs, as this may have severe consequences for the choice of the 'levels' of the Multi-level Monte-Carlo method.