2,051 research outputs found
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
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