Regularity of multifractional moving average processes with random Hurst exponent

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Abstract

A recently proposed alternative to multifractional Brownian motion (mBm) with random Hurst exponent is studied, which we refer to as Itô-mBm. It is shown that Itô-mBm is locally self-similar. In contrast to mBm, its pathwise regularity is almost unaffected by the roughness of the functional Hurst parameter. The pathwise properties are established via a new polynomial moment condition similar to the Kolmogorov–Centsov theorem, allowing for random local Hölder exponents. Our results are applicable to a broad class of moving average processes where pathwise regularity and long memory properties may be decoupled, e.g. to a multifractional generalization of the Matérn process.

Introduction

Regularity or roughness of the sample paths of a stochastic process Yt,t0, is a crucial property, e.g., when studying models defined by stochastic integrals with respect to (w.r.t.) Yt. It can be measured in terms of the (local) p-variation, a non-increasing function of 0<p<, and paths with infinite 1-variation are called rough. Properties and the related calculus of rough functions is reviewed and discussed in [23]. Alternatively, one may establish Hölder-continuity and make use of the fact that α-Hölder continuity implies bounded 1/α-variation. A convenient criterion is the Kolmogorov–Centsov Theorem (see e.g. [16, Thm. 2.8]), which links Hölder-regularity of the paths to absolute moments of the increments. This well-known result can be used to establish Hölder coefficients which (i) hold uniformly on the interval of interest and (ii) are deterministic. This is satisfactory for most conventional processes of interest, e.g. fractional Brownian motion or solutions of stochastic differential equations. For general stochastic processes, the regularity of sample paths neither needs to be deterministic nor globally constant. A very simple example is the stopped Brownian motion (Wtτ)t0 for a random stopping time τ>0. On [0,τ), the process is (almost) 1/2 Hölder continuous, whereas it is arbitrarily smooth on (τ,) as a constant. Hence, the smoothness at any fixed time t is random.

The process primarily considered in this paper is closely related to the class of multifractional Brownian motion (mBm), which generalizes fractional Brownian motion (fBm) by allowing the Hurst exponent to vary in time. Multifractional Brownian motions have been used as a model for stochastic volatility in finance [11], for network traffic [8], and for temperature time series [20], among others. As there exist various definitions of mBm in the literature leading to slightly different processes [7], [18], [25], [29], we shall confine our discussion to a special case of [29, Def. 1.1]. Let Wt be a Brownian motion defined on a filtered probability space (Ω,(Ft)tR,F,P), and define B(t,H)=t(ts)+H12(s)+H12dWs,tR,H(0,1),which is a random field indexed by t and H. For fixed H, B(t,H) is a fBm with parameter H. For a deterministic process Ht, the mBm may be defined as BtH=B(t,Ht), i.e. BtH=B(t,Ht)=t(ts)+Ht12(s)+Ht12dWs,tR.Intuitively, BtH behaves like a fractional Brownian motion with Hurst parameter Ht locally around t.

To describe the local regularity of a stochastic process such as BtH, we define for a generic process Yt the pointwise Hölder coefficient as αt(Y)=supα:lim supε0sup|h|<ε|Yt+hYt||h|α=0.It has been shown by [14, Prop. 13] that for any t, almost surely, αt(BH)=Htαt(H).That is, αt(BH) is a modification of Htαt(H), but the latter two processes are in general not indistinguishable [2].

The identity (4) for the pointwise Hölder exponent reveals that the regularity of the functional Hurst exponent Ht itself significantly affects the smoothness of BtH. In the extreme case that Ht is a discontinuous step-function, we readily find from (2) that BtH itself is discontinuous. In particular, tB(t,H) is continuous for all H(0,1), but B(t,H)B(t,H) for HH because the integrands differ. This discontinuity invalidates the intuition that BtH should behave locally like a fBm with Hurst parameter Ht. The relevance of the regularity of Ht even extends to the statistical estimation of Ht for a fixed t. For example, the rates of convergence of the nonparametric estimators of [6] are vacuous unless αt(H)>Ht (see Proposition 3 therein). While nonparametric estimators always require some smoothness of the target quantity, i.e. αt(H), the relevance of the difference αt(H)Ht is a special feature of inference for mBm. When estimating the global Hölder regularity inftHt, [19] also require that Ht is η-Hölder continuous for some η>suptHt.

Another shortcoming of defining a multifractional process in terms of (2) is that this construction cannot easily be extended to a stochastic (Ft-adapted) process Ht. In fact, if Ht in (2) is random, the integrand is in general no longer Fs-adapted and the stochastic integral cannot be interpreted in the Itô sense. A possible remedy is to either assume Ht to be independent of the driving Gaussian noise (Wt)tR, or to employ an alternative representation of the random field B(t,H) [5].

As an alternative to overcome the latter measurability issues, [4] suggest to instead study the process,1 KtH=tσs(ts)+Hs12(s)+Hs12dWs.By making the integrand depend on Hs instead of Ht, the integral in (5) can be defined in the classical Itô sense. A similar process has also been proposed by [30], though for a deterministic Hurst function Hs. The process KtH has been called a multifractional process with random exponent (MPRE) by [4]. To contrast this process with mBm, we will refer to KtH as an Itô-mBm. To study the regularity of KtH, [4] derive a series representation based on a wavelet basis, which can be investigated via analytical methods. Their findings suggest that the smoothness of (5) is less sensitive to the regularity of Ht. In particular, under the condition that inftHt>1/2 and that Ht is η-Hölder continuous for η>1/2, they find that the Hölder regularity of KtH on any interval [t1,t2] is at least infs[t1,t2]Hs. Although the applicability of this result is restricted, it allows for the case η<Ht, much unlike (4).

The regularity results derived by [4] are based on a wavelet representation of KtH. In contrast, the novel Kolmogorov–Centsov type result presented in Section 2 allows us to simplify the analysis of the Itô-mBm by probabilistic methods, and we are able to obtain refined results on its pathwise regularity. In particular, we show in Section 3 that its pointwise Hölder exponent satisfies αt(KH)Ht if Ht is continuous, irrespective of the regularity of Ht. Different from the analysis of [4], we may include the cases Ht12 and αt(H)12. In contrast to the classical mBm BtH, the paths of Ht may be rougher than those of the corresponding Itô-mBm KtH. For the case that Ht is discontinuous, we also obtain similar results which are only slightly weaker. Notably, in the latter case, KtH is still continuous, in contrast to the process BtH.

To show that our estimates of the pointwise Hölder exponent are sharp, i.e. that αt(KH)=Ht, we establish a rescaling limit showing that, for each fixed t, as h0, hHt(Kt+hrHKtH)σtr(rs)+Ht12(s)+Ht12dW̃s,for a Brownian motion W̃s independent of Ht. That is, the process KtH behaves locally near t like a fractional Brownian motion with Hurst parameter Ht. The precise limit theorem is presented in Section 3. All our results hold for a broader class of moving average processes which contains, for example, a multifractional generalization of the Mátern process, see Section 3.

The processes KtH and BtH are in general different, except for the trivial case where Ht is constant. Nevertheless, both processes may serve as a non-stationary generalization of fractional Brownian motion. In particular, the scaling relation (6) also holds for BtH if Ht is sufficiently smooth, since HB(t,H) is C [2]. Thus, the presented smoothness results for KtH could be an argument to use the latter process in practice.

This paper is structured as follows. In Section 2, we present the continuity criterion result for random fields with random, local Hölder exponents. These results are applied in Section 3 to study the smoothness of the Itô-mBm KtH. All proofs are gathered in Appendix A.

For two real numbers a,b, we denote ab=min(a,b), ab=max(a,b), and (a)+=max(a,0). To make clear where we study random fields, we denote scalar indices s,tR by normal letters and vector indices s,tRd by bold letters. We denote by C a generic constant, the value of which might change from line to line. If the factor C depends on the quantities a,b,c, we denote this as C(a,b,c). Weak convergence in metric spaces is denoted as . For an interval IR, the space of continuous functions equipped with the supremum norm is denoted by C(I). Convergence in probability is denoted as P, and for a sequence of random variables ɛn, we write ɛn=oP(1) if ɛnP0 as n.

Section snippets

Pathwise Hölder continuity of random fields

On a probability space (Ω,F,P), we consider a real-valued random field (Yt)t indexed by t[0,T]d, i.e. each Yt is a random variable. The Kolmogorov–Centsov theorem requires that there are two positive real numbers α,β>0, and a constant C, such that for all s,t[0,T]d, it holds E|YsYt|αCstd+β,where is an arbitrary norm on Rd. If (7) holds, there exists a modification Ỹt of Yt, i.e. P(Ỹt=Yt)=1, such that the paths of Ỹt are η-Hölder continuous for any η(0,β/α). While this is a result

A multifractional Gaussian process

The aim of this section is to apply the results of the previous section to the Itô-mBm KtH employing a random but adapted functional Hurst exponent and to study some properties of the process. But before proceeding, let us recall some notions. A stationary process has long memory, if its correlations are not integrable. If its correlation function, c(), exhibits a power-law behavior at infinity, c(τ)|τ|γ, for some γ(0,1), the process has long memory with Hurst exponent H=1γ/2. Thus, for 1/2

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank the anonymous reviewer for his careful reading and constructive comments which helped to improve this article considerably.

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