Structure-Preserving Integrators for Lévy-Driven Stochastic Systems

The fundamental aim of this proposal is to further the understanding of stochastic differential equations driven by Lévy processes and their algebraic structures, the design and analysis of novel structure-preserving integrators when the system is constrained to evolve on a manifold and the modelling of such systems.

In many applications the evolution of quantities is random in nature. Key stochastic processes for describing the random driving force are Wiener processes as models for Gaussian random noise, and more generally Lévy processes, as generalizations of Wiener processes in applications when randomness cannot always be captured accurately by Gaussian random factors. Imagine zooming into a time series of financial data by increasing the frequency of observations. In econometric studies it has been observed that on the smaller time scale data will typically exhibit larger fluctuations and hence non-Gaussian behaviour. With the increasing amount of data we are now able to observe and to process, more complex stochastic differential equations driven by Lévy processes are becoming increasingly more important. Applications are numerous, including in climate science, where changes in some weather patterns have been observed to occur in jumps, in the modelling of financial quantities such as stock prices and interest rates, or in biology for example in models for the movement of cells.

It is typical that stochastic differential equations describing these evolutions, even in the continuous case, have no known solution as a given function of the driving stochastic processes. The design and analysis of numerical integrators is thus pivotal in modelling and in understanding and analysing these equations. This challenge is compounded when the solution to the stochastic differential system is known to evolve on a manifold, and even further when dealing with the jump discontinuities of a Lévy process. Standard Taylor series-based numerical integrators are not designed to generate approximate solutions that remain on the manifold, and projecting these approximate solutions onto the manifold may not be feasible or efficient.

The proposed research programme aims to address this challenge. It is based on recent findings of the proposer and collaborators that link in an intrinsic way stochastic differential equations and their integrators with algebraic structures that encompass key properties of the stochastic system under consideration and that enable the design of novel efficient integrators that are more accurate than standard Taylor series expansion schemes in Gaussian- and Lévy-driven stochastic differential equations and the design of structure-preserving methods for continuous stochastic differential equations.

The current research project aims to extend these methods to stochastic differential equations driven by Lévy processes. It will bring together ideas from stochastic analysis, stochastic differential geometry, algebra, and numerical analysis. The project will further the understanding of stochastic differential equations driven by Lévy processes and evolving on manifolds and the intrinsic link to their algebraic structure, and it will develop novel generic structure-preserving numerical methods for solving Lévy-driven models that were previously not solvable.