Regularity of multifractional moving average processes with random Hurst exponent
Introduction
Regularity or roughness of the sample paths of a stochastic process , is a crucial property, e.g., when studying models defined by stochastic integrals with respect to (w.r.t.) . It can be measured in terms of the (local) -variation, a non-increasing function of , and paths with infinite -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 -Hölder continuity implies bounded -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 for a random stopping time . On , the process is (almost) Hölder continuous, whereas it is arbitrarily smooth on as a constant. Hence, the smoothness at any fixed time 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 be a Brownian motion defined on a filtered probability space , and define which is a random field indexed by and . For fixed , is a fBm with parameter . For a deterministic process , the mBm may be defined as , i.e. Intuitively, behaves like a fractional Brownian motion with Hurst parameter locally around .
To describe the local regularity of a stochastic process such as , we define for a generic process the pointwise Hölder coefficient as It has been shown by [14, Prop. 13] that for any , almost surely, That is, is a modification of , 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 itself significantly affects the smoothness of . In the extreme case that is a discontinuous step-function, we readily find from (2) that itself is discontinuous. In particular, is continuous for all , but for because the integrands differ. This discontinuity invalidates the intuition that should behave locally like a fBm with Hurst parameter . The relevance of the regularity of even extends to the statistical estimation of for a fixed . For example, the rates of convergence of the nonparametric estimators of [6] are vacuous unless (see Proposition 3 therein). While nonparametric estimators always require some smoothness of the target quantity, i.e. , the relevance of the difference is a special feature of inference for mBm. When estimating the global Hölder regularity , [19] also require that is -Hölder continuous for some .
Another shortcoming of defining a multifractional process in terms of (2) is that this construction cannot easily be extended to a stochastic (-adapted) process . In fact, if in (2) is random, the integrand is in general no longer -adapted and the stochastic integral cannot be interpreted in the Itô sense. A possible remedy is to either assume to be independent of the driving Gaussian noise , or to employ an alternative representation of the random field [5].
As an alternative to overcome the latter measurability issues, [4] suggest to instead study the process,1 By making the integrand depend on instead of , 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 . The process has been called a multifractional process with random exponent (MPRE) by [4]. To contrast this process with mBm, we will refer to as an Itô-mBm. To study the regularity of , [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 . In particular, under the condition that and that is -Hölder continuous for , they find that the Hölder regularity of on any interval is at least . Although the applicability of this result is restricted, it allows for the case , much unlike (4).
The regularity results derived by [4] are based on a wavelet representation of . 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 if is continuous, irrespective of the regularity of . Different from the analysis of [4], we may include the cases and . In contrast to the classical mBm , the paths of may be rougher than those of the corresponding Itô-mBm . For the case that is discontinuous, we also obtain similar results which are only slightly weaker. Notably, in the latter case, is still continuous, in contrast to the process .
To show that our estimates of the pointwise Hölder exponent are sharp, i.e. that , we establish a rescaling limit showing that, for each fixed , as , for a Brownian motion independent of . That is, the process behaves locally near like a fractional Brownian motion with Hurst parameter . 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 and are in general different, except for the trivial case where 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 if is sufficiently smooth, since is [2]. Thus, the presented smoothness results for 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 . All proofs are gathered in Appendix A.
For two real numbers , we denote , , and . To make clear where we study random fields, we denote scalar indices by normal letters and vector indices by bold letters. We denote by a generic constant, the value of which might change from line to line. If the factor depends on the quantities , we denote this as . Weak convergence in metric spaces is denoted as . For an interval , the space of continuous functions equipped with the supremum norm is denoted by . Convergence in probability is denoted as , and for a sequence of random variables , we write if as .
Section snippets
Pathwise Hölder continuity of random fields
On a probability space , we consider a real-valued random field indexed by , i.e. each is a random variable. The Kolmogorov–Centsov theorem requires that there are two positive real numbers , and a constant , such that for all , it holds where is an arbitrary norm on . If (7) holds, there exists a modification of , i.e. , such that the paths of are -Hölder continuous for any . 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 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, , exhibits a power-law behavior at infinity, , for some , the process has long memory with Hurst exponent . Thus, for
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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2022, Stochastic Processes and their ApplicationsCitation Excerpt :Among many other things, the following theorem has been obtained in [17]. Let us point out that the keystone of the proofs of Theorems 1.1 and 1.2 given in [17] is the important Burkholder–Davis–Gundy inequality (see for instance [18,19]) as formulated in the following proposition: It is worth mentioning that a straightforward consequence of Theorems 1.1 and 1.5 and (1.1) is that: