Statistics of Lévy-driven Models
Final Report Abstract
Time series models defined in continuous time are important for stochastic modelling in many areas of applications like e.g. finance, insurance, physics, signal processing and control theory. To obtain realistic marginal distributions and dynamics and to ensure a high mathematical tractability Lévy processes - a class of stochastic processes including, for instance, Brownian motion, Poisson processes and alpha-stable Lévy motions - are used as the random driving force. In this project we considered (multivariate) models which are of moving average type and especially the very important class of continuous-time autoregressive moving average (CARMA) processes. We considerably advanced the understanding of their statistical and probabilistic properties and developed a concise theory of statistical inference for them assuming that the processes are observed only at finitely many points in time. Thus, several estimators for (multivariate) stationary Lévy-driven moving average and CARMA processes were defined and analysed. In particular, we went beyond standard assumptions and considered very heavily-tailed cases (without finite variance), long range dependent cases when the Rosenblatt distribution appeared as a limit distribution, estimators robust to outliers as well as non-equidistantly sampled data. Likewise, we considered not only the estimation of the parameters when the autoregressive and moving average order of a CARMA process is known, but also how to estimate these orders appropriately using information criteria. Frequently Fourier techniques were employed and estimation carried out in the frequency domain. Since often multivariate data sets are not stationary, but certain linear ´ combinations are stationary, we investigated the definition of co-integration in a multivariate Levydriven CARMA framework, the properties of the arising processes and developed in full detail the related statistical inference. We also studied Lévy-driven (mixed) moving average processes which generalize CARMA processes and allow for more flexibility, as they are, for instance, also capable of exhibiting long range dependence. For example, for the popular supOU (stochastic volatility) model we obtained consistent moment based estimators. In general we obtained important insight in the behaviour of their sample autocorrelation function, their periodogram, their dependence structure and their extremal behaviour. As in many applications one is confronted with data that appears stationary in the short run, but whose dynamics (slowly) changes in the long run, we started to introduce the notion of local stationarity for continuous-time moving average processes and to investigate when e.g. CARMA processes with time varying coefficients belong to this new class. To improve the applicability of our estimators we studied various bootstrap methods, in particular for certain linear processes observed at low frequency and for Lévy driven continuous time autoregressive processes, making advantage of the specific parametric structure of the processes under consideration.
Publications
- (2012). Quasi maximum likelihood estimation for strongly mixing state space models and multivariate CARMA processes. Electron. J. Stat. 6, 2185–2234
Schlemm, E. and Stelzer, R.
- (2013). A central limit theorem for the sample autocorrelations of a Lévy driven continuous time moving average process. J. Statist. Plann. Inference 143, 1295–1306
Cohen, S. and Lindner, A.
(See online at https://doi.org/10.1016/j.jspi.2013.03.022) - (2013). Functional regular variation of Lévy-driven multivariate mixed moving average processes. Extremes 16, 351–382
Moser, M. and Stelzer, R.
(See online at https://doi.org/10.1007/s10687-012-0165-y) - (2013). Integration of CARMA processes and spot volatility modelling. J. Time Series Anal. 34, 156–167
Brockwell, P. J. and Lindner, A.
(See online at https://doi.org/10.1111/jtsa.12011) - (2013). Noise recovery for Lévy-driven CARMA processes and high-frequency behaviour of approximating Riemann sums. Electron. J. Stat. 7, 533–561
Ferrazzano, V. and Fuchs, F.
(See online at https://doi.org/10.1214/13-EJS783) - (2013). On the limit behavior of the periodogram of highfrequency sampled stable CARMA processes. Stochastic Process. Appl 121, 1, 229–273
Fasen, V. and Fuchs, F.
(See online at https://doi.org/10.1016/j.spa.2012.08.003) - (2013). Spectral estimates for high-frequency sampled CARMA processes. J. Time Series Anal. 34, 532–551
Fasen, V. and Fuchs, F.
- (2013). Spectral representation of multivariate regularly varying Lévy and CARMA processes. J. Theoret. Probab 26, 2, 410–436
Fuchs, F. and Stelzer, R.
(See online at https://doi.org/10.1007/s10959-011-0369-0) - (2014). Asymptotics for autocovariances and integrated periodograms for linear processes observed at lower frequencies. International Statistical Review 82, 123–140
Niebuhr, T. and Kreiss, J.-P.
(See online at https://doi.org/10.1111/insr.12019) - (2014). Bootstrapping continuoustime autoregressive processes. Ann. Inst. Statist. Math. 66, 75–92
Brockwell, P. J., Kreiss, J.-P., and Niebuhr, T.
(See online at https://doi.org/10.1007/s10463-013-0406-0) - (2014). Limit theory for high frequency sampled MCARMA models. Adv. Appl. Probab. 46, 846–877
Fasen, V.
(See online at https://doi.org/10.1239/aap/1409319563) - (2015). Moment based estimation of supOU processes and a related stochastic volatility model. Stat. Risk Model. 32, 1, 1–24
Stelzer, R., Tosstorff, T., and Wittlinger, M.
(See online at https://doi.org/10.1515/strm-2012-1152) - (2015). Prediction of Lévy driven CARMA processes. J. Econometrics 189, 263–271
Brockwell, P. J. and Lindner, A.
(See online at https://doi.org/10.1016/j.jeconom.2015.03.021) - (2016). Dependence estimation for high frequency sampled multivariate CARMA models. Scand. J. Statist. 43, 292–320
Fasen, V.
(See online at https://doi.org/10.1111/sjos.12180) - (2016). Information criteria for multivariate CARMA processes. Bernoulli
Fasen, V. and Kimmig, S.
