Lévy driven CARMA generalized processes and stochastic partial differential equations
Introduction
Autoregressive moving average (ARMA) processes are very well known processes in time series analysis. An ARMA process , , is given by where are deterministic coefficients and is white noise or even an independent and identically distributed (iid) sequence of random variables. In short form we can also write where , are polynomials and is the shift operator defined by for . ARMA processes were generalized in various ways and have many applications, e.g. in finance, astrophysics, engineering and traffic data, see [5], [9], [22] and [13].
As the solution of (1.1) is a discrete process on a lattice, a possible way to generalize the concept is to study a continuous version of (1.1), which is called continuous ARMA (CARMA) process. A CARMA process , where , is given by where is a -valued process satisfying the stochastic differential equation with where are deterministic coefficients such that and for every , denotes the transpose of and is a two-sided Lévy process. Eqs. (1.2), (1.3) are the so called state-space representation of the formal stochastic differential equation with the differential operator and and are polynomials. In [17] necessary and sufficient conditions on and were given such that there exists a strictly stationary solution of (1.2), (1.3), namely it was shown that it is sufficient and necessary that . CARMA processes have many applications, see [7] and [2].
As the CARMA process is defined on , spatial problems cannot be easily incorporated. As a consequence, there are some extensions of the CARMA process to the multidimensional setting. Lately, there were the two papers of Brockwell and Matsuda [4] and Pham [16], who introduce different concepts of CARMA processes in the multidimensional setting. In [4] the new CARMA random field was given by where denotes the integration over a Lévy basis, and are polynomials such that and some further restrictions. The model has a well understood second order behavior and can be used for statistical estimation. However, the authors do not deal with a dynamical description.
Pham [16] follows another way and defines a CARMA random field as a mild solution of the system of SPDEs given by where is a Lévy basis, are matrices and is the identity matrix. Pham speaks of causal CARMA random fields, as the solution of the system (1.5) depends only on the past in the sense that the solution at point depends solely on the behavior of on . So we can see directly that there is a big difference between these two definitions.
The aim of this paper is to find a connection between these two models and give a generalized definition of CARMA random fields. Our starting point is the equation where are polynomials in variables, denotes the differential operator and denotes Lévy white noise. Our solution is defined as a generalized solution, see Section 3. We will start with an abstract analysis of this problem and prove for a far more general class than (1.7) the existence of a generalized solution under relatively mild conditions on the Lévy white noise. Our solution is similar to the definition of generalized CARMA process in [3] and as there, we do not assume that the degree of the polynomial is higher than the degree of the polynomial . We will discuss two examples, which are related to the processes of Brockwell and Matsuda [4] and Pham [16]. We will also give certain conditions on and that guarantee that the obtained generalized solutions are random fields.
The above mentioned results can be found in Sections 3 SPDEs and generalized solutions, 4 CARMA generalized processes, where our main results are Theorem 3.4, Theorem 4.3. In Section 2 we recall some basic notation. In Section 3 we recall the definitions of Lévy white noise and generalized random processes. Moreover, we prove that a convolution operator with certain properties regarding its integrability defines a generalized random process and as an application we will study stochastic homogeneous elliptic partial differential equations. In Section 4 we use Theorem 3.4 to show the existence of our CARMA generalized processes. Moreover, we study the concept of mild solutions in Section 5, prove existence of mild CARMA random fields and show some connections between the mild and generalized solutions. In Section 6 we study the moment properties of our CARMA random fields and show that if the Lévy white noise has existing -moment for some , then the CARMA random field has also finite -moment, see Proposition 6.1. In Section 7 we will study the connection between our model and the CARMA random field of Brockwell and Matsuda [4].
Section snippets
Notation and preliminaries
To fix notation, by we denote a measurable space, where is a set and is a -algebra and by we denote all measurable functions with respect to where . In the case that and are clear from the context we set . If we consider a probability space , where is a probability measure on , we say that a sequence converges to in if converges in probability to with respect to the measure . In the case of
The concept of generalized solutions
This section deals with Lévy white noise and the definition of solutions of the SPDEs given in (1.7). We will prove a multiplier theorem for general Lévy white noise and use this theorem to prove the existence of our CARMA random process. We will follow mainly [6, Section 2].
As already mentioned, we denote by the space of infinitely differentiable functions with compact support, where we assume that the space is equipped with the usual topology, i.e. we say that a sequence
CARMA generalized processes
We construct a generalization of CARMA processes. A CARMA generalized process is a generalized solution of a special SPDE.
Definition 4.1 Let be a Lévy white noise, and be polynomials of the form and A generalized process is called a CARMA generalized process if solves the equation which means that
Recall that and . For classical CARMA processes in dimension 1
CARMA random fields
Until now we have only studied generalized solutions of the CARMA SPDE (1.7), but in the vast literature of stochastic partial differential equations driven by Lévy noise the concept of mild solutions seems to be more used, as the mild solution is itself a random field. We show under stronger conditions the existence of a mild solution of (1.7). But first we recall what a mild solution is.
Definition 5.1 Let and be partial differential operators and let be a locally integrable fundamentalSee [21]
Moment properties
We say that a generalized process has existing -moment, , if for every .
Let be a Lévy white noise with characteristic triplet . Then it is easy to see (cf. [20, Theorem 25.3, p. 159]) that has existing -moment if and only if Next we show that if has existing -moment then so has the CARMA generalized process given in Theorem 4.3.
Proposition 6.1 Let have existing -moment () and let and be polynomials satisfying Assumption 4.2. Then
CARMA random fields in the sense of Brockwell and Matsuda
We will now analyze the CARMA random fields in the sense of Brockwell and Matsuda defined in [4] and show that the corresponding random field defines a mild solution of a fractional stochastic partial differential equation. In our setting we find for odd dimensions the corresponding CARMA generalized processes with respect to a SPDE of type (4.1). A CARMA random field in the sense of Brockwell and Matsuda is defined as follows: Let , be a polynomial with
Acknowledgment
Partial support by DFG grant LI 1026/6-1 is gratefully acknowledged. The author is grateful to the referees for their valuable suggestions, which helped to improve the paper greatly. The author would like to thank Alexander Lindner for his patience and support and for many interesting and fruitful discussions. Moreover, the author would like to thank Claudia Klüppelberg and Viet Son Pham for a valuable discussion on CARMA random fields. Last but not least the author would like to thank Paul
References (22)
- et al.
CARMA(p,q) generalized random processes
J. Stat. Plan. Inference
(2010) - et al.
Prediction of functional ARMA processes with an application to traffic data
Econ. Stat.
(2017) - et al.
Quasi Ornstein–Uhlenbeck processes
Bernoulli
(2011) - et al.
High-frequency sampling and kernel estimation for continuous-time moving average processes
J. Time Ser. Anal.
(2013) - et al.
Continuous auto-regressive moving average random fields on
J. R. Stat. Soc. B
(2016) - S. Eyheramendy, F. Elorrieta, W. Palma, An autoregressive model for irregular time series of variable stars, in:...
- et al.
Unified view on Lévy white noise: general integrability conditions and applications to linear SPDE
(2018) - et al.
Estimation of stable CARMA models with an application to electricitiy spot prices
Stat. Modell.
(2011) - et al.
Generalized Functions 4: Applications of Harmonic Analysis
(1964) - et al.
Financial applications of ARMA models with GARCH errors
J. Risk Finance
(2006)
Classical Fourier Analysis
Cited by (11)
The domain of definition of the Lévy white noise
2021, Stochastic Processes and their ApplicationsCitation Excerpt :Being the probabilistic adaptation of the theory of generalized functions of L. Schwartz, generalized random processes are very flexible, and many finite-dimensional results of probability theory have natural extension to this infinite-dimensional setting [10,19,34,43]. Generalized random processes have been used as a natural framework for CARMA random processes [17] and CARMA random fields [7,8], for the scaling limits of statistical models in quantum field theory [1,2,61], where the continuous-domain limit fields are often too irregular to admit a pointwise representation [18,20,35]. One of the advantages of the theory of generalized random processes is that it allows to properly define random processes which do not admit a pointwise representation.
Some Singularities of Linear AR Processes Characterization in Applied Problems of Power Equipment and Power Systems Diagnosis
2024, Studies in Systems, Decision and ControlSome features of the systems for monitoring and diagnostic hydro units technical condition with considering smart grid technology
2023, IOP Conference Series: Earth and Environmental ScienceAlmost periodic stationary processes
2022, arXivCENTRAL LIMIT THEOREMS FOR STATIONARY RANDOM FIELDS UNDER WEAK DEPENDENCE WITH APPLICATION TO AMBIT AND MIXED MOVING AVERAGE FIELDS
2022, Annals of Applied Probability