Continuous-time ARMA Processes Research Articles

Empirical spectral processes for stationary state space models

In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving Lévy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving Lévy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the Lévy process is weaker. Furthermore, we show the weak convergence in both the space of continuous functions and in the dual space to a Gaussian process and give an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander–Rosenblatt statistic and the Cramér–von Mises statistic. We present the exact limit distributions of both statistics and show their performance through a simulation study.

On nonnegative solutions of SDDEs with an application to CARMA processes

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best of our knowledge, no simple nonnegativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\ge 3$. Finally, we extend the result to a multidimensional setting.

Stochastic delay differential equations and related autoregressive models

ABSTRACTIn this paper we suggest two continuous-time models which exhibit an autoregressive structure. We obtain existence and uniqueness results and study the structure of the solution processes. One of the models, which corresponds to general stochastic delay differential equations, will be given particular attention. We use the obtained results to link the introduced processes to both discrete-time and continuous-time ARMA processes.

A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Levy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same Levy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p−1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Levy process, and show simulations and applications to real data.

Prediction of Lévy-driven CARMA processes

The conditional expectations, E(Y(h)|Y(u),−∞<u≤0) and E(Y(h)|Y(u),−M≤u≤0) with h>0 and 0<M<∞ are determined for a continuous-time ARMA (CARMA) process (Y(t))t∈R driven by a Lévy process L with E|L(1)|<∞. If E(L(1)2)<∞ these are the minimum mean-squared error predictors of Y(h) given (Y(t))t≤0 and (Y(t))−M≤t≤0 respectively. Conditions are also established under which the sample-path of L can be recovered from that of Y, both when Y is causal and strictly stationary and (without these assumptions) when L is a pure-jump Lévy process. When E(L(1)2)<∞ and Y is causal and strictly stationary the best linear predictors P(Y(h)|Y(u),u≤0) and P(Y(h)|Y(−nΔ),n∈N) are also determined, the latter yielding a simple algorithm for determining the parameters of the ARMA process obtained by sampling the CARMA process at regular intervals.

A Lévy-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 Lévy 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 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.

Embedding a Gaussian discrete‐time autoregressive moving average process in a Gaussian continuous‐time autoregressive moving average process

Abstract. Embedding a discrete‐time autoregressive moving average (DARMA) process in a continuous‐time ARMA (CARMA) process has been discussed by many authors. These authors have considered the relationship between the autocovariance structures of continuous‐time and related discrete‐time processes. In this article, we treat the problem from a slightly different point of view. We define embedding in a more rigid way by taking account of the probability structure. We consider Gaussian processes. First we summarize the necessary and sufficient condition for a DARMA process to be able to be embedded in a CARMA process. Secondly, we show a concrete condition such that a DARMA process can be embeddable in a CARMA process. This condition is new and general. Thirdly, we show some special cases including new examples. We show how we can examine embeddability for these special cases.

Multivariate fractionally integrated CARMA processes

A multivariate analogue of the fractionally integrated continuous time autoregressive moving average (FICARMA) process defined by Brockwell [Representations of continuous-time ARMA processes, J. Appl. Probab. 41 (A) (2004) 375–382] is introduced. We show that the multivariate FICARMA process has two kernel representations: as an integral over the fractionally integrated CARMA kernel with respect to a Lévy process and as an integral over the original (not fractionally integrated) CARMA kernel with respect to the corresponding fractional Lévy process (FLP). In order to obtain the latter representation we extend FLPs to the multivariate setting. In particular we give a spectral representation of FLPs and consequently, derive a spectral representation for FICARMA processes. Moreover, various probabilistic properties of the multivariate FICARMA process are discussed. As an example we consider multivariate fractionally integrated Ornstein–Uhlenbeck processes.

USING CONTINUOUS-TIME MODELING FOR ERRORS-IN-VARIABLES IDENTIFICATION

Continuous-time identification is applied to an errors-in-variables setting. A continuous-time model is fitted to data consisting of discrete-time noise corrupted input and output measurements. The noise-free input is modelled as a continuous-time ARMA process. It is described how the Cramér-Rao lower bound for the estimation problem can be computed. Several parameter estimation approaches for the problem are presented, and also illustrated in a short numerical study.

On possibilities for estimating continuous-time ARMA parameters

The problem of estimating the parameters in continuous-time ARMA processes from discrete-time data is considered. Three different approaches, based on the prediction error method, the instrumental variable method and an approximate maximum likelihood method, respectively, arc studied. All three techniques provide reliable solutions to the estimation problem. A general discussion of the inherent difficulties of the problem is given together with an extensive numerical study.

Lévy-driven CARMA processes

Properties and examples of continuous-time ARMA (CARMA) processes driven by Levy processes are examined. By allowing Levy processes to replace Brownian motion in the definition of a Gaussian CARMA process, we obtain a much richer class of possibly heavy-tailed continuous-time stationary processes with many potential applications in finance, where such heavy tails are frequently observed in practice. If the Levy process has finite second moments, the correlation structure of the CARMA process is the same as that of a corresponding Gaussian CARMA process. In this paper we make use of the properties of general Levy processes to investigate CARMA processes driven by Levy processes {W(t)} without the restriction to finite second moments. We assume only that W (1) has finite r-th absolute moment for some strictly positive r. The processes so obtained include CARMA processes with marginal symmetric stable distributions.

Discrete and continuous time cointegration

We give a definition of integration and cointegration in continuous time and study characterizations of those definitions, in particular for CARMA (Continuous time ARMA) processes. We check also that such singularities in CARMA processes may imply unit roots problems in the discretized corresponding process. Then, error correction representations in continuous time are exhibited and discussed, and a general theorem of representation giving a precise description of the singularities is proved. Lastly, we look at other singularities emphasizing the importance of estimating the continuous time model if it is the true one: a one-dimensional noise can generate a two-dimensional regular AR process.

On the parameters estimation of continuous-time ARMA processes from noisy observations

Recently, an iterative algorithm has been presented for estimating the parameters of partially observed continuous-time processes [1]. In this note we concentrate on continuous-time ARMA processes observed in white noise. A maximum a-posteriori (MAP) estimator is defined for the trajectory of the parameters' random process. This approach enables the MAP estimation of randomly slowly varying parameters, and extends the conventional treatment of time-invariant parameters. The iterative algorithm derived for the MAP estimation, increases the posterior probability of the parameters in each iteration, and converges to a stationary point of the posterior probability functional. Each iteration involves a standard linear smoother followed by a finite-dimensional linear system, and thus is easily implemented.