Cylindrical Levy Processes and Their Applications

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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.