High-dimensional statistics for point and jump processes

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Whenever the evolution of a time dependent system is influenced by random phenomena, stochastic processes are used for mathematical modelling. The applications are manifold appearing in natural sciences as well as in engineering, econometrics and financial mathematics. The calibration of these probabilistic models is a fundamental requirement for their application and has attracted much attention in the last decades. While, except for a few examples, most of the statistical research for stochastic processes is restricted to one-dimensional or low-dimensional models, an important feature of data sets in modern applications is high dimensionality, i.e., the dimension of the parameter space or dimension of the process may increase with the sample size. Examples include a huge number of covariates which potentially influence the evolution of biochemical processes as well as large portfolios in financial markets. It is well known that classical procedures fail if the dimension is large and various novel methods of high-dimensional parameter estimation have been developed. The aim of this project is to combine the statistical theory for stochastic processes with high-dimensional statistics to construct and analyse new statistical methods for high-dimensional stochastic processes.