Cylindrical Levy Processes and Their Applications

Stochastic differential equations model a process evolving in time and subject to a random noise. Numerous phenomena in nature and economics are modelled by these equations. The reason for the random noise might be found in external or internal fluctuations which do not allow a deterministic description, in random events in the future or in uncertainty of the model. The complexity of the model, e.g. the numbers of parameters involved or the state space of the modelled process, often results in the necessity to consider stochastic differential equations in infinite dimensional spaces. However, up to now, most of these models are restricted to a continuous Gaussian noise and to infinite dimensional spaces with a very rich structure due to the lack of a satisfactory mathematical theory.The first objective of this project is to develop a theory which enables us to treat stochastic differential equations in infinite dimensional spaces of a general type. The random source might have discontinuous paths and is allowed to be of a very general form, such that the randomness not only depends on the evolution in time but also on the underlying space. In the second part of this project, the usability of the theory is verified by studying two concrete examples out of the numerous applications: one model describes the physical distribution of the heat in a given region subject to some external random noise, and the second model originates from financial mathematics and describes the evolution of interest rate curves.

The outputs of this project have the potential to establish more realistic models with a better accordance to observed data, in turn impacting the numerous fields where stochastic differential equations in infinite dimensions are applied. Consequently, the outcomes of the project can be expected to have an impact not just within academia but also in the economy, industry and government where these models are applied.An important part of the project is the development of software for sampling methods and numerical approximations for the forward curves and prices of interest rate derivatives in a significantly improved model. The algorithms will be provided to the financial package em Premia, which has been used by a consortium of banks, including Credit Agricole, Societe Generale and Bank Austria. After two years the software becomes available on Premia's web site and can be freely downloaded for academic purposes.In modern areas of financial mathematics and related fields, such as real option theory, climatology and resource exploitation, the complexity of the models seem to be captured better by stochastic differential equations in infinite dimensional spaces. An important part of the proposed project is to explore the possibility of improving and of understanding the models in these and other non-standard areas by using an infinite dimensional approach. For this purpose, the PI will initiate several cooperations with groups working in these areas and the Centre for Interdisciplinary Computational and Dynamical Analysis at the University of Manchester along with its industrial contacts.