Stationary and multi-self-similar random fields with stochastic volatility

This paper introduces stationary and multi-self-similar random fields which account for stochastic volatility and have type G marginal law. The stationary random fields are constructed using volatility modulated mixed moving average fields and their probabilistic properties are discussed. Also, two methods for parameterising the weight functions in the moving average representation are presented: One method is based on Fourier techniques and aims at reproducing a given correlation structure, the other method is based on ideas from stochastic partial differential equations. Moreover, using a generalised Lamperti transform we construct volatility modulated multi-self-similar random fields which have type G distribution

A weak law of large numbers for estimating the correlation in bivariate Brownian semistationary processes

This article presents various weak laws of large numbers for the so-called realised covariation of a bivariate stationary stochastic process which is not a semimartingale. More precisely, we consider two cases: Bivariate moving average processes with stochastic correlation and bivariate Brownian semistationary processes with stochastic correlation. In both cases, we can show that the (possibly scaled) realised covariation converges to the integrated (possibly volatility modulated) stochastic correlation process

Limit theorems for multivariate Brownian semistationary processes and feasible results

In this paper we introduce the \textit{multivariate} Brownian semistationary (BSS) processes and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general stationary multivariate Gaussian processes, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.Comment: 35 page

Likelihood theory for the Graph Ornstein-Uhlenbeck process

We consider the problem of modelling restricted interactions between continuously-observed time series as given by a known static graph (or network) structure. For this purpose, we define a parametric multivariate Graph Ornstein-Uhlenbeck (GrOU) process driven by a general L\'evy process to study the momentum and network effects amongst nodes, effects that quantify the impact of a node on itself and that of its neighbours, respectively. We derive the maximum likelihood estimators (MLEs) and their usual properties (existence, uniqueness and efficiency) along with their asymptotic normality and consistency. Additionally, an Adaptive Lasso approach, or a penalised likelihood scheme, infers both the graph structure along with the GrOU parameters concurrently and is shown to satisfy similar properties. Finally, we show that the asymptotic theory extends to the case when stochastic volatility modulation of the driving L\'evy process is considered

A central limit theorem for the realised covariation of a bivariate Brownian semistationary process

This article presents a weak law of large numbers and a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process. The novelty of our results lies in the fact that we derive the suitable asymptotic theory both in a multivariate setting and outside the classical semimartingale framework. The proofs rely heavily on recent developments in Malliavin calculus

Bridging between short-range and long-range dependence with mixed spatio-temporal Ornstein-Uhlenbeck processes

While short-range dependence is widely assumed in the literature for its simplicity, long-range dependence is a feature that has been observed in data from finance, hydrology, geophysics and economics. In this paper, we extend a L\'evy-driven spatio-temporal Ornstein-Uhlenbeck process by randomly varying its rate parameter to model both short-range and long-range dependence. This particular set-up allows for non-separable spatio-temporal correlations which are desirable for real applications, as well as flexible spatial covariances which arise from the shapes of influence regions. Theoretical properties such as spatio-temporal stationarity and second-order moments are established. An isotropic g-class is also used to illustrate how the memory of the process is related to the probability distribution of the rate parameter. We develop a simulation algorithm for the compound Poisson case which can be used to approximate other L\'evy bases. The generalised method of moments is used for inference and simulation experiments are conducted with a view towards asymptotic properties

Scoring Predictions at Extreme Quantiles

Prediction of quantiles at extreme tails is of interest in numerous applications. Extreme value modelling provides various competing predictors for this point prediction problem. A common method of assessment of a set of competing predictors is to evaluate their predictive performance in a given situation. However, due to the extreme nature of this inference problem, it can be possible that the predicted quantiles are not seen in the historical records, particularly when the sample size is small. This situation poses a problem to the validation of the prediction with its realisation. In this article, we propose two non-parametric scoring approaches to assess extreme quantile estimates. These methods are based on predicting a sequence of equally extremal quantiles on different parts of the data. We then use the quantile scoring function to evaluate the competing predictors. The performance of the scoring methods is compared with the conventional scoring method and superiority of the former methods are demonstrated in a simulation study. The methods are then applied to reanalyse cyber Netflow data from Los Alamos National Laboratory and daily precipitation data at a station in California available from Global Historical Climatology Network

A L\'evy-driven rainfall model with applications to futures pricing

We propose a parsimonious stochastic model for characterising the distributional and temporal properties of rainfall. The model is based on an integrated Ornstein-Uhlenbeck process driven by the Hougaard L\'evy process. We derive properties of this process and propose an extended model which generalises the Ornstein-Uhlenbeck process to the class of continuous-time ARMA (CARMA) processes. The model is illustrated by fitting it to empirical rainfall data on both daily and hourly time scales. It is shown that the model is sufficiently flexible to capture important features of the rainfall process across locations and time scales. Finally we study an application to the pricing of rainfall derivatives which introduces the market price of risk via the Esscher transform. We first give a result specifying the risk-neutral expectation of a general moving average process. Then we illustrate the pricing method by calculating futures prices based on empirical daily rainfall data, where the rainfall process is specified by our model.Comment: 28 pages, 9 figure

A latent trawl process model for extreme values

This paper presents a new model for characterising temporal dependence in exceedances above a threshold. The model is based on the class of trawl processes, which are stationary, infinitely divisible stochastic processes. The model for extreme values is constructed by embedding a trawl process in a hierarchical framework, which ensures that the marginal distribution is generalised Pareto, as expected from classical extreme value theory. We also consider a modified version of this model that works with a wider class of generalised Pareto distributions, and has the advantage of separating marginal and temporal dependence properties. The model is illustrated by applications to environmental time series, and it is shown that the model offers considerable flexibility in capturing the dependence structure of extreme value data

Hybrid simulation scheme for volatility modulated moving average fields

We develop a simulation scheme for a class of spatial stochastic processes called volatility modulated moving averages. A characteristic feature of this model is that the behaviour of the moving average kernel at zero governs the roughness of realisations, whereas its behaviour away from zero determines the global properties of the process, such as long range dependence. Our simulation scheme takes this into account and approximates the moving average kernel by a power function around zero and by a step function elsewhere. For this type of approach the authors of [8], who considered an analogous model in one dimension, coined the expression hybrid simulation scheme. We derive the asymptotic mean square error of the simulation scheme and compare it in a simulation study with several other simulation techniques and exemplify its favourable performance in a simulation study