45 results found
Passeggeri R, Veraart A, 2019, Mixing properties of multivariate infinitely divisible random fields, Journal of Theoretical Probability, Vol: 32, Pages: 1845-1879, ISSN: 0894-9840
In this work we present different results concerning mixing properties of multivariate infinitely divis-ible (ID) stationary random fields. First, we derive some necessary and sufficient conditions for mixingof stationary ID multivariate random fields in terms of their spectral representation. Second, we provethat (linear combinations of independent) mixed moving average fields are mixing. Further, using a sim-ple modification of the proofs of our results we are able to obtain weak mixing versions of our results.Finally, we prove the equivalence of ergodicity and weak mixing for multivariate ID stationary randomfields.
Heinrich C, Pakkanen MS, Veraart AED, 2019, Hybrid simulation scheme for volatility modulated moving average fields, Mathematics and Computers in Simulation, Vol: 166, Pages: 224-244, ISSN: 0378-4754
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 , 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.
Passeggeri R, Veraart A, 2019, Limit theorems for multivariate Brownian semistationary processes and feasible results, Advances in Applied Probability, Vol: 51, Pages: 667-716, ISSN: 0001-8678
In this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, 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.
Granelli A, Veraart A, A central limit theorem for the realised covariation of a bivariate Brownian semistationary process, Bernoulli, ISSN: 1350-7265
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.
Veraart A, 2019, Modeling, simulation and inference for multivariate time series of counts using trawl processes, Journal of Multivariate Analysis, Vol: 169, Pages: 110-129, ISSN: 0047-259X
This article presents a new continuous-time modeling framework for multivariate time series of counts which have an infinitely divisible marginal distribution. The model is based on a mixed moving average process driven by Lévy noise, called a trawl process, where the serial correlation and the cross-sectional dependence are modeled independently of each other. Such processes can exhibit short or long memory. We derive a stochastic simulation algorithm and a statistical inference method for such processes. The new methodology is then applied to high frequency financial data, where we investigate the relationship between the number of limit order submissions and deletions in a limit order book.
Deschatre T, Veraart A, 2018, A JOINT MODEL FOR ELECTRICITY SPOT PRICES AND WINDPENETRATION WITH DEPENDENCE IN THE EXTREMES, Forecasting and risk management for renewable energy, Editors: Drobinski, Mougeot, Picard, Plougonven, Tankov
Noven R, Veraart A, Gandy A, 2018, A latent trawl process model for extreme values, Journal of Energy Markets, Vol: 11, Pages: 1-24, ISSN: 1756-3607
This paper presents a new model for characterising temporaldependence in exceedancesabove 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 byembedding a trawl process in a hierarchical framework, which ensures that the marginaldistribution is generalised Pareto, as expected from classical extreme value theory. Wealso consider a modified version of this model that works witha wider class of generalisedPareto distributions, and has the advantage of separating marginal and temporal depen-dence properties. The model is illustrated by applicationsto environmental time series,and it is shown that the model offers considerable flexibilityin capturing the dependencestructure of extreme value data
Nguyen M, Veraart A, 2018, Bridging between short-range and long-range dependence with mixed spatio-temporal Ornstein-Uhlenbeck processes, Stochastics: An International Journal of Probability and Stochastic Processes, Vol: 90, Pages: 1023-1052, ISSN: 1744-2508
While short-range dependence is widely assumed in the literature for its simplicity, long-range dependence is a featurethat has been observed in data from finance, hydrology, geophysics and economics. In this paper, we extend a L´evy-drivenspatio-temporal Ornstein-Uhlenbeck process by randomly varying its rate parameter to model both short-range and longrangedependence. This particular set-up allows for non-separable spatio-temporal correlations which are desirable forreal applications, as well as flexible spatial covariances which arise from the shapes of influence regions. Theoreticalproperties such as spatio-temporal stationarity and second-order moments are established. An isotropic g-class is alsoused to illustrate how the memory of the process is related to the probability distribution of the rate parameter. Wedevelop a simulation algorithm for the compound Poisson case which can be used to approximate other L´evy bases. Thegeneralised method of moments is used for inference and simulation experiments are conducted with a view towardsasymptotic properties.
Granelli A, Veraart A, 2017, A central limit theorem for the realised covariation of a bivariate Brownian semistationary process
Granelli A, Veraart A, 2017, A weak law of large numbers for estimating the correlation in bivariate Brownian semistationary processes
Veraart AED, 2017, Essentials of Probability Theory for Statisticians, JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION, Vol: 112, Pages: 879-880, ISSN: 0162-1459
Nguyen M, Veraart A, 2017, Modelling spatial heteroskedasticity by volatility modulated moving averages, Spatial Statistics, Vol: 20, Pages: 148-190, ISSN: 2211-6753
Spatial heteroskedasticity has been observed in many spatial data applications such as air pollution and vegetation. We propose a model, the volatility modulated moving average, to account for changing variances across space. This stochastic process is driven by Gaussian noise and involves a stochastic volatility field. It is conditionally non-stationary but unconditionally stationary: a useful property for theory and practice. We develop a discrete convolution algorithm as well as a two-step moments-matching estimation method for simulation and inference respectively. These are tested via simulation experiments and the consistency of the estimators is proved under suitable double asymptotics. To illustrate the advantages that this model has over the usual Gaussian moving average or process convolution, sea surface temperature anomaly data from the International Research Institute for Climate and Society are analysed.
Veraart A, Book review of "Essentials of Probability Theory for Statisticians" by Michael A. Proschan and Pamela A. Shaw, Journal of the American Statistical Association, ISSN: 1537-274X
Nguyen M, Veraart A, 2016, Spatio-temporal Ornstein-Uhlenbeck processes: theory, simulation and statistical inference, Scandinavian Journal of Statistics, Vol: 44, Pages: 46-80, ISSN: 1467-9469
Spatio-temporal modelling is an increasingly popular topic in Statistics. Our paper contributes to this line of researchby developing the theory, simulation and inference for a spatio-temporal Ornstein-Uhlenbeck process. We conduct detailedsimulation studies and demonstrate the practical relevance of these processes in an empirical study of radiationanomaly data. Finally, we describe how predictions can be carried out in the Gaussian setting.
Sauri O, Veraart A, On the class of distributions of subordinated Lévy processes, Stochastic Processes and Their Applications, ISSN: 0304-4149
This article studies the class of distributions obtained by subordinating L´evyprocesses and L´evy bases by independent subordinators and meta-times. To dothis we derive properties of a suitable mapping obtained via L´evy mixing. Weshow that our results can be used to solve the so-called recovery problem forgeneral L´evy bases as well as for moving average processes which are driven bysubordinated L´evy processes.
Granelli A, Veraart A, 2016, Modelling the variance risk premium of equity indices: the role ofdependence and contagion, SIAM Journal on Financial Mathematics, Vol: 7, Pages: 382-417, ISSN: 1945-497X
The variance risk premium (VRP) refers to the premium demanded for holding assetswhose variance is exposed to stochastic shocks.This paper identifies a new modelling framework for equity indices and presents for thefirst time explicit analytical formulas for their VRP in a multivariate stochastic volatilitysetting, which includes multivariate non-Gaussian Ornstein-Uhlenbeck processes and Wishartprocesses. Moreover, we propose to incorporate contagion within the equity index via amultivariate Hawkes process and find that the resulting dynamics of the VRP represent aconvincing alternative to the models studied in the literature up to date. We show that ournew model can explain the key stylised facts of both equity indices and individual assets andtheir corresponding VRP, while some popular (multivariate) stochastic volatility models mayfail.
Veraart A, Zdanowicz H, 2015, Modelling and predicting photovoltaic power generation in the EEX market, SSRN
Sauri O, Veraart A, 2015, On the class of distributions of subordinated Levy processes
Nguyen M, Veraart A, 2015, Tempo-spatial Ornstein-Uhlenbeck processes: theory, simulation and statistical inference
Veraart AED, 2015, Modelling the impact of wind power production on electricity prices by regime-switching Levy semistationary processes, Stochastics of Environmental and Financial Economics, Editors: Benth, Di Nunno, Publisher: Springer, Pages: 321-340
This paper studies the impact of wind power production on electricity prices in the European energy market.We propose a new modelling framework based on so-called regime-switching Levy semistationary processes to account for forward-looking information consisting of predicted wind power generation. We show that our new regime-switching model, where the regime switch depends on the so-called wind penetration index, can describe recent electricity price data very well.
Barndorff-Nielsen OE, Benth FE, Veraart AED, 2015, Cross-commodity modelling by multivariate ambit fields, Commodities, Energy and Environmental Finance, Editors: Aid, Ludkovski, Sircar, Publisher: Springer, Pages: 109-148
Barndorff--Nielsen OE, Benth FE, Veraart AED, 2014, Modelling Electricity Futures by Ambit Fields, Advances of Applied Probability, Vol: 46, Pages: 719-745
Noven RC, Veraart AED, Gandy A, 2014, A Levy-driven rainfall model with applications to futures pricing
Barndorff-Nielsen OE, Benth FE, Veraart AED, 2014, Recent advances in ambit stochastics with a view towards tempo-spatial stochastic volatility/intermittency, Banach Center Publications, Vol: 104, Pages: 25-60
Barndorff-Nielsen OE, Lunde A, Shephard N, et al., 2014, Integer-valued trawl processes: A class of stationary infinitely divisible processes, Scandinavian Journal of Statistics, Vol: 41, Pages: 693-724
Granelli A, Veraart AED, 2014, Modelling the Variance Risk Premium of Equity Indices: The Role of Dependence and Contagion
Benth FE, Eyjolfsson H, Veraart AED, 2014, Approximating Levy semistationary processes via Fourier methods in the context of power markets, SIAM Journal on Financial Mathematics, Pages: 71-98
Veraart AED, Veraart LAM, 2013, Risk premiums in energy markets, Journal of Energy Markets, Pages: 91-132
Veraart AED, 2013, Stationary and multi-self-similar random fields with stochastic volatility, Stochastics, Vol: 87, Pages: 848-870, ISSN: 0090-9491
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 (MA) fields and their probabilistic properties are discussed. Also, two methods for parameterizing the weightfunctions in the MA 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 generalized Lamperti transform we construct volatility modulated multi-self-similar random fields which have type G distribution.
Barndorff-Nielsen OE, Benth FE, Pedersen J, et al., 2013, On stochastic integration for volatility modulated Levy-driven Volterra processes, Stochastic Processes and Their Applications, Vol: n/a, ISSN: 0304-4149
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