High order splitting methods for SDEs

In this talk, we will discuss how ideas from rough path theory can be leveraged to develop high order numerical methods for SDEs. To motivate our approach, we consider what happens when the Brownian motion driving an SDE is replaced by a piecewise linear path. We show that this procedure transforms the SDE into a sequence of ODEs – which can then be discretized using an appropriate ODE solver. Moreover, to achieve a high accuracy, we construct these piecewise linear paths to match certain features of the Brownian motion. At the same time, the ODEs obtained from this path-based approach can be interpreted as stages in a splitting method, which neatly connects our work to the existing literature. We give several examples to demonstrate the flexibility and convergence properties of this methodology. 

However, extending high order methods to general SDEs is difficult as certain non-Gaussian iterated integrals of Brownian motion must be considered. In fact, the exact simulation of such integrals (or equivalent Lévy areas) is still an open problem. To address this challenge, we propose the use of machine learning and train a neutral network model to generate Lévy area samples. In our experiments, we observe that our “LévyGAN” model generates high-quality samples of Lévy area in a fraction of the time that it takes traditional approaches. Finally, we demonstrate how variance reduction can be achieved for a LévyGAN-powered splitting method using Multilevel Monte Carlo.

 Joint with Gonçalo dos Reis, Calum Strange, Andraž Jelinčič, Jiajie Tao, William Turner, Thomas Cass and Hao Ni.

The talk will be followed by refreshments in the senior common room (level 5, Huxley building) at 4pm.