Non-stationary processes and their application to financial high-frequency data

The thesis is devoted to non-stationary point process models as generalizations of the standard homogeneous Poisson process. The work can be divided in two parts. In the first part, we introduce a fractional non-homogeneous Poisson process (FNPP) by applying a random time change to the standard Poisson process. We characterize the FNPP by deriving its non-local governing equation. We further compute moments and covariance of the process and discuss the distribution of the arrival times. Moreover, we give both finite-dimensional and functional limit theorems for the FNPP and the corresponding fractional non-homogeneous compound Poisson process. The limit theorems are derived by using martingale methods, regular variation properties and Anscombe's theorem. Eventually, some of the limit results are verified via a Monte-Carlo simulation. In the second part, we analyze statistical point process models for durations between trades recorded in financial high-frequency trading data. We consider parameter settings for models which are non-stationary or very close to non-stationarity which is quite typical for estimated parameter sets of models fitted to financial data. Simulation, parameter estimation and in particular model selection are discussed for the following three models: a non-homogeneous normal compound Poisson process, the exponential autoregressive conditional duration model (ACD) and a Hawkes process model. In a Monte-Carlo simulation, we test the performance of the following information criteria for model selection: Akaike's information criterion, the Bayesian information criterion and the Hannan-Quinn information criterion. We are particularly interested in the relation between the rate of correct model selection and the underlying sample size. Our numerical results show that the model selection for the compound Poisson type model works best for small parameter numbers. Moreover, the results for Hawkes processes confirm the theoretical asymptotic distributions of model selection whereas for the ACD model the model selection exhibits adverse behavior in certain cases.