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The aim of this paper is to develop estimation and inference methods for the drift parameters of multivariate L\'evy-driven continuous-time autoregressive processes of order $p\in\mathbb{N}$. Starting from a continuous-time observation of the process, we develop consistent and asymptotically normal maximum likelihood estimators. We then relax the unrealistic assumption of continuous-time observation by considering natural discretizations based on a combination of Riemann-sum, finite difference, and thresholding approximations. The resulting estimators are also proven to be consistent and asymptotically normal under a general set of conditions, allowing for both finite and infinite jump activity in the driving L\'evy process. When discretizing the estimators, allowing for irregularly spaced observations is of great practical importance. In this respect, CAR($p$) models are not just relevant for "true" continuous-time processes: a CAR($p$) specification provides a natural continuous-time interpolation for modeling irregularly spaced data - even if the observed process is inherently discrete. As a practically relevant application, we consider the setting where the multivariate observation is known to possess a graphical structure. We refer to such a process as GrCAR and discuss the corresponding drift estimators and their properties. The finite sample behavior of all theoretical asymptotic results is empirically assessed by extensive simulation experiments.

Uncertainty can be classified as either aleatoric (intrinsic randomness) or epistemic (imperfect knowledge of parameters). The majority of frameworks assessing infectious disease risk consider only epistemic uncertainty. We only ever observe a single epidemic, and therefore cannot empirically determine aleatoric uncertainty. Here, we characterise both epistemic and aleatoric uncertainty using a time-varying general branching process. Our framework explicitly decomposes aleatoric variance into mechanistic components, quantifying the contribution to uncertainty produced by each factor in the epidemic process, and how these contributions vary over time. The aleatoric variance of an outbreak is itself a renewal equation where past variance affects future variance. We find that, superspreading is not necessary for substantial uncertainty, and profound variation in outbreak size can occur even without overdispersion in the offspring distribution (i.e. the distribution of the number of secondary infections an infected person produces). Aleatoric forecasting uncertainty grows dynamically and rapidly, and so forecasting using only epistemic uncertainty is a significant underestimate. Therefore, failure to account for aleatoric uncertainty will ensure that policymakers are misled about the substantially higher true extent of potential risk. We demonstrate our method, and the extent to which potential risk is underestimated, using two historical examples.

We develop a GMM approach for estimation of log-normal stochastic volatility models driven by a fractional Brownian motion with unrestricted Hurst exponent. We show that a parameter estimator based on the integrated variance is consistent and, under stronger conditions, asymptotically normally distributed. We inspect the behavior of our procedure when integrated variance is replaced with a noisy measure of volatility calculated from discrete high-frequency data. The realized estimator contains sampling error, which skews the fractal coefficient toward "illusive roughness." We construct an analytical approach to control the impact of measurement error without introducing nuisance parameters. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show the bias correction attenuates any systematic deviance in the parameter estimates. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in stochastic volatility., Comment: 52 pages, 5 figures, v4: title of previous version in footnote corrected

This article studies the finite sample behaviour of a number of estimators for the integrated power volatility process of a Brownian semistationary process in the non semi-martingale setting. We establish three consistent feasible estimators for the integrated volatility, two derived from parametric methods and one non-parametrically. We then use a simulation study to compare the convergence properties of the estimators to one another, and to a benchmark of an infeasible estimator. We further establish bounds for the asymptotic variance of the infeasible estimator and assess whether a central limit theorem which holds for the infeasible estimator can be translated into a feasible limit theorem for the non-parametric estimator., Comment: 21 pages, 7 figures

We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method and in simulations we observe that the bootstrap-based hypothesis test provides considerable finite-sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data; we illustrate this by applying the bootstrap method to two empirical data sets: we assess the roughness of a time series of high-frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.

We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency when the data of interest are generated by a non-semimartingale, or a Brownian semistationary process in particular. This estimation method is motivated by the assessment of relative energy dissipation in empirical data of turbulence, but it is also applicable in other areas. We develop a probabilistic asymptotic theory for realised relative power variations of Brownian semistationary processes, and introduce inference methods based on the theory. We also discuss how to extend the asymptotic theory to other classes of processes exhibiting stochastic volatility/intermittency. As an empirical application, we study relative energy dissipation in data of atmospheric turbulence., Comment: 25 pages, 4 figures, v3: major revision, this version contains an application to electricity prices that was omitted from the published version

This paper presents some asymptotic results for statistics of Brownian semi-stationary (BSS) processes. More precisely, we consider power variations of BSS processes, which are based on high frequency (possibly higher order) differences of the BSS model. We review the limit theory discussed in [Barndorff-Nielsen, O.E., J.M. Corcuera and M. Podolskij (2011): Multipower variation for Brownian semistationary processes. Bernoulli 17(4), 1159-1194; Barndorff-Nielsen, O.E., J.M. Corcuera and M. Podolskij (2012): Limit theorems for functionals of higher order differences of Brownian semi-stationary processes. In "Prokhorov and Contemporary Probability Theory", Springer.] and present some new connections to fractional diffusion models. We apply our probabilistic results to construct a family of estimators for the smoothness parameter of the BSS process. In this context we develop estimates with gaps, which allow to obtain a valid central limit theorem for the critical region. Finally, we apply our statistical theory to turbulence data., Comment: 25 pages, 2 figures