## Final Report Summary - TSSV (Tempo-spatial stochastic volatility: Modelling and statistical inference)

Statistics for space-time data is one of the most important research frontiers in modern statistics. This project proposed to introduce and develop the concept of space-time stochastic volatility, which allows one to model volatility clusters both in time and in space. Empirical evidence for stochastic volatility is ubiquitous, and hence, it is vital and urgent that statistical models and estimation methods will be developed to account for this key quantity. Stochastic volatility is a latent variable, meaning that it is not directly observable but needs to be estimated from other (observable) variables.

Key research objectives include the development of suitable classes of stochastic models for space-time volatility which can realistically describe random fluctuations and clusters in volatility depending on its corresponding spatial and temporal location. In addition, efficient inference methods need to be found so that the new models can be fitted to empirical data and can be used by practitioners.

The project has been concluded successfully and the key results obtained can be divided into contributions to stochastic modelling and to stochastic simulation and inference.

On the modelling side, a reliable base model for spatio-temporal phenomena has been developed, which consists of a moving average process driven by an infinitely divisible independently scattered random measure. This process is parameterised by a one-dimensional time and a multidimensional space parameter. Relevant parametric classes of such models have been developed using two key ideas: One approach is based on the idea of linking the corresponding base model to stochastic partial differential equations and using (extended) Green’s functions to parameterise the weight function in the moving average representation. The other approach is driven by the idea of finding a suitable weight function which results in a given spatio-temporal autocorrelation function. This can be done using suitable Fourier inversion techniques applied to the autocorrelation function.

In a next step, it has been shown how spatio-temporal stochastic volatility can be added to such a base model: Either volatility modulation can be achieved through stochastic scaling, meaning that stochastic volatility enters in the corresponding stochastic integral as a multiplicative factor in the integrand, or volatility can be added through stochastic time- and space-transformation and/or extended subordination. Suitable parametric models for both forms of volatility modulation have been developed in this project. In particular, spatio-temporal Ornstein-Uhlenbeck processes and tools for stochastic simulation and statistical inference for such processes have been developped. Also, volatility modulated moving average processes have been developed and simulation and inference methods are now available. These new classes of stochastic processes and the suitable simulation and inference methods have been tested in extensive simulation studies.

On the inference side, the following results have been obtained so far: The base model proposed in the project can be reliably estimated using the generalised method of moments and quasi-maximumlikelihood methods. In addition, if observations in space are sparse, but rather frequent in time, it might be more appropriate to tackle the concept of space-time volatility in a multivariate time-series setting where the individual components represent different spatial locations. Space-time volatility then translates to stochastic (co)-volatility between the various components. In this project, we focused on modelling such multivariate time series by a class of continuous-time stochastic processes called Brownian semistationary processes, which are moving average processes driven by a Brownian motion which is scaled by a stochastic volatility factor. As such, such processes are generally not Gaussian and not semimartingales either; this complicates their analysis. In the bivariate setting, a weak law of large numbers and a central limit theorem has been derived to estimate the (possibly stochastic) covariation between the two components. This is an important step towards a full analysis of space-time volatility in a setting for sparse spatial, but frequent temporal observations.

Also, the problem of incorporating stochastic volatility into a Levy-driven moving average process by means of extended subordination has been studied and answers to the question under which conditions we can identify the law of the extended subordinator given observations from the corresponding moving average process are now available.

In addition to the theoretical results, the new models have been successfully applied in empirical studies ranging from modelling precipitation, air pollution, radiation anomaly, sea surface temperature anomalies to renewable energy production data and high frequency financial data. As such the theoretical and empirical results are of key relevance to the Work Programme since it strongly contributes to the initiative “sustainable growth”: The results of this research project are directly applicable to, e.g. measuring and modelling the risk associated with climate change and to finding an optimal design for wind farms, making renewable sources of energy more efficient and reliable. It is expected that the ongoing dissemination activities will further increase the socio-economic impact of the completed research.

The research grant has enabled the fellow to fully integrate into the European research community and to obtain a senior and permanent position at the host university. During the fellowship the research fellow has graduated four PhD students and their current research group consists now of four PhD students. Many new interesting research collaborations have arisen from the research project and follow-up research is already ongoing.

Key research objectives include the development of suitable classes of stochastic models for space-time volatility which can realistically describe random fluctuations and clusters in volatility depending on its corresponding spatial and temporal location. In addition, efficient inference methods need to be found so that the new models can be fitted to empirical data and can be used by practitioners.

The project has been concluded successfully and the key results obtained can be divided into contributions to stochastic modelling and to stochastic simulation and inference.

On the modelling side, a reliable base model for spatio-temporal phenomena has been developed, which consists of a moving average process driven by an infinitely divisible independently scattered random measure. This process is parameterised by a one-dimensional time and a multidimensional space parameter. Relevant parametric classes of such models have been developed using two key ideas: One approach is based on the idea of linking the corresponding base model to stochastic partial differential equations and using (extended) Green’s functions to parameterise the weight function in the moving average representation. The other approach is driven by the idea of finding a suitable weight function which results in a given spatio-temporal autocorrelation function. This can be done using suitable Fourier inversion techniques applied to the autocorrelation function.

In a next step, it has been shown how spatio-temporal stochastic volatility can be added to such a base model: Either volatility modulation can be achieved through stochastic scaling, meaning that stochastic volatility enters in the corresponding stochastic integral as a multiplicative factor in the integrand, or volatility can be added through stochastic time- and space-transformation and/or extended subordination. Suitable parametric models for both forms of volatility modulation have been developed in this project. In particular, spatio-temporal Ornstein-Uhlenbeck processes and tools for stochastic simulation and statistical inference for such processes have been developped. Also, volatility modulated moving average processes have been developed and simulation and inference methods are now available. These new classes of stochastic processes and the suitable simulation and inference methods have been tested in extensive simulation studies.

On the inference side, the following results have been obtained so far: The base model proposed in the project can be reliably estimated using the generalised method of moments and quasi-maximumlikelihood methods. In addition, if observations in space are sparse, but rather frequent in time, it might be more appropriate to tackle the concept of space-time volatility in a multivariate time-series setting where the individual components represent different spatial locations. Space-time volatility then translates to stochastic (co)-volatility between the various components. In this project, we focused on modelling such multivariate time series by a class of continuous-time stochastic processes called Brownian semistationary processes, which are moving average processes driven by a Brownian motion which is scaled by a stochastic volatility factor. As such, such processes are generally not Gaussian and not semimartingales either; this complicates their analysis. In the bivariate setting, a weak law of large numbers and a central limit theorem has been derived to estimate the (possibly stochastic) covariation between the two components. This is an important step towards a full analysis of space-time volatility in a setting for sparse spatial, but frequent temporal observations.

Also, the problem of incorporating stochastic volatility into a Levy-driven moving average process by means of extended subordination has been studied and answers to the question under which conditions we can identify the law of the extended subordinator given observations from the corresponding moving average process are now available.

In addition to the theoretical results, the new models have been successfully applied in empirical studies ranging from modelling precipitation, air pollution, radiation anomaly, sea surface temperature anomalies to renewable energy production data and high frequency financial data. As such the theoretical and empirical results are of key relevance to the Work Programme since it strongly contributes to the initiative “sustainable growth”: The results of this research project are directly applicable to, e.g. measuring and modelling the risk associated with climate change and to finding an optimal design for wind farms, making renewable sources of energy more efficient and reliable. It is expected that the ongoing dissemination activities will further increase the socio-economic impact of the completed research.

The research grant has enabled the fellow to fully integrate into the European research community and to obtain a senior and permanent position at the host university. During the fellowship the research fellow has graduated four PhD students and their current research group consists now of four PhD students. Many new interesting research collaborations have arisen from the research project and follow-up research is already ongoing.