Lévy driven CARMA generalized processes and stochastic partial differential equations

https://doi.org/10.1016/j.spa.2020.04.009Get rights and content

Abstract

We give a new definition of a Lévy driven CARMA random field, defining it as a generalized solution of a stochastic partial differential equation (SPDE). Furthermore, we give sufficient conditions for the existence of a mild solution of our SPDE. Our model unifies all known definitions of CARMA random fields, and in particular for dimension 1 we obtain the classical CARMA process.

Introduction

Autoregressive moving average (ARMA) processes are very well known processes in time series analysis. An ARMA(p,q) process (Xk)kZ, p,qN0, is given by Xki=1paiXki=Wk+j=1qbjWkj,where a1,,ap,b1,,bq are deterministic coefficients and (Wk)kZ is white noise or even an independent and identically distributed (iid) sequence of random variables. In short form we can also write a(B)Xk=b(B)Wk,where a(z)=1i=1paizk, b(z)=1+j=1qbjzj are polynomials and B is the shift operator defined by BlYk=Ykl for lN. ARMA(p,q) 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(p,q) process (Xt)tR, where p>q, is given by Xt=bYt,tR,where Y=(Yt)tR is a p-valued process satisfying the stochastic differential equation dYt=AYtdt+epdLtwith A=010000100001apap1ap2a1,ep=0001p and b=b0b1bp2bp1,where a1,,ap,b0,,bp1 are deterministic coefficients such that bq0 and bj=0 for every j>q, b denotes the transpose of b and L=(Lt)tR is a two-sided Lévy process. Eqs. (1.2), (1.3) are the so called state-space representation of the formal stochastic differential equation a(D)Yt=b(D)DLt,with D the differential operator and a(z)=zp+a1zp1++ap and b(z)=b0+b1z++bqzq are polynomials. In [17] necessary and sufficient conditions on L and A 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 Elog+(|L1|)<. CARMA processes have many applications, see [7] and [2].

As the CARMA process is defined on R, 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 Sd(t)Rdr=1pb(λr)a(λr)eλrtudL(u),where dL denotes the integration over a Lévy basis, a and b are polynomials such that a(z)=i=1p(z2λi2) 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 Y as a mild solution of the system of SPDEs given by Y(t)=bX(t),tRd,(IpdAd)(Ip1A1)X(t)=cL̇(t),tRd, where L̇ is a Lévy basis, A1,,AdRp×p are matrices and Ip 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 x depends solely on the behavior of L̇ on (,x1]××(,xd]. 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 p(D)s=q(D)L̇,where p,q are polynomials in d variables, D denotes the differential operator and L̇ denotes Lévy white noise. Our solution s 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(p,q) process in [3] and as there, we do not assume that the degree of the polynomial p is higher than the degree of the polynomial q. 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 p and q 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 α-moment for some 0<α2, then the CARMA random field has also finite α-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 (Ω,F) we denote a measurable space, where Ω is a set and F is a σ-algebra and by L0(Ω,F,K) we denote all measurable functions f:ΩK with respect to F where K=R,. In the case that F and K are clear from the context we set L0(Ω)=L0(Ω,F,K). If we consider a probability space (Ω,F,P), where P is a probability measure on (Ω,F), we say that a sequence (fn)nNL0(Ω) converges to f in L0(Ω) if fn converges in probability to f with respect to the measure P. In the case of (Rd,B(Rd))

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 D(Rd) 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 (φn)nND(Rd)

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 L̇ be a Lévy white noise, n,mN0 and p,q:RdR be polynomials of the form p(x)=|α|npαxαand q(x)=|α|mqαxα.A generalized process s:D(Rd)L0(Ω) is called a CARMA(p,q) generalized process if s solves the equation p(D)s=q(D)L̇,which means that s,p(D)φ=L̇,q(D)φ a.s. for every φD(Rd).

Recall that p(D)=p(D) and q(D)=q(D). 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

See [21]

Let p(D) and q(D) be partial differential operators and let G:RdR be a locally integrable fundamental

Moment properties

We say that a generalized process s:DL0(Ω) has existing β-moment, β>0, if E|s,φ|β< for every φD(Rd).

Let L̇ be a Lévy white noise with characteristic triplet (a,γ,ν). Then it is easy to see (cf. [20, Theorem 25.3, p. 159]) that L̇ has existing β-moment if and only if |z|>1|z|βν(dz)<.Next we show that if L̇ has existing β-moment then so has the CARMA generalized process given in Theorem 4.3.

Proposition 6.1

Let L̇ have existing β-moment (β>0) and let p and q 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 0q<p, a(z)=zp+a1zp1++ap=i=1p(zλi) 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)

  • BrockwellP.J. et al.

    CARMA(p,q) generalized random processes

    J. Stat. Plan. Inference

    (2010)
  • KlepschJ. et al.

    Prediction of functional ARMA processes with an application to traffic data

    Econ. Stat.

    (2017)
  • Barndorff-NielsenO.E. et al.

    Quasi Ornstein–Uhlenbeck processes

    Bernoulli

    (2011)
  • BrockwellP.J. et al.

    High-frequency sampling and kernel estimation for continuous-time moving average processes

    J. Time Ser. Anal.

    (2013)
  • BrockwellP.J. et al.

    Continuous auto-regressive moving average random fields on Rn

    J. R. Stat. Soc. B

    (2016)
  • S. Eyheramendy, F. Elorrieta, W. Palma, An autoregressive model for irregular time series of variable stars, in:...
  • FageotJ. et al.

    Unified view on Lévy white noise: general integrability conditions and applications to linear SPDE

    (2018)
  • GarciaI. et al.

    Estimation of stable CARMA models with an application to electricitiy spot prices

    Stat. Modell.

    (2011)
  • GelfI.M. et al.

    Generalized Functions 4: Applications of Harmonic Analysis

    (1964)
  • GhahramaniM. et al.

    Financial applications of ARMA models with GARCH errors

    J. Risk Finance

    (2006)
  • GrafakosL.

    Classical Fourier Analysis

    (2008)
  • Cited by (11)

    View all citing articles on Scopus
    View full text