Assessing Relative Volatility/Intermittency/Energy Dissipation

We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency when the data of interest are generated by a non-semimartingale, or a Brownian semistationary process in particular. This estimation method is motivated by the assessment of relative energy dissipation in empirical data of turbulence, but it is also applicable in other areas. We develop a probabilistic asymptotic theory for realised relative power variations of Brownian semistationary processes, and introduce inference methods based on the theory. We also discuss how to extend the asymptotic theory to other classes of processes exhibiting stochastic volatility/intermittency. As an empirical application, we study relative energy dissipation in data of atmospheric turbulence.


Introduction
The concept of (stochastic) volatility is of central importance in many fields of science. In some of these the term intermittency is used instead of volatility. Thus volatility/intermittency has a central role in mathematical finance and financial econometrics (Barndorff-Nielsen and Shephard, 2010), in turbulence, rain and cloud studies (Lovejoy and Schertzer, 2006;Waymire, 2006) and other aspects of environmental science (Pichard and Abbott, 2012), in relation to nanoscale emitters (Frantsuzov et al., 2013), magnetohydrodynamics (Mininni and Pouquet, 2009), and to liquid mixtures of chemicals (Sreenivasan, 2004), and last but not least in the physics of fusion plasmas (Carbone et al., 2000). In turbulence the key concept of energy dissipation is subsumed under that of intermittency. For discussions of intermittency and energy dissipation in turbulence see Frisch (1995, Chapter 6) and Tsinober (2009, Chapter 7) (cf. also the illustration on p. 20 of the latter reference).
Speaking generally, volatility/intermittency is taken to mean that the phenomenon under study exhibits more variation than expected; that is, more than the most basic type of random influence (often thought of as Gaussian) envisaged. Hence volatility/intermittency is a relative concept, and its meaning depends on the particular setting under investigation. Once that meaning is clarified the question is how to assess the volatility/intermittency empirically and then to describe it in stochastic terms and incorporate it in a suitable probabilistic model.
Such 'additional' random fluctuations generally vary, in time and/or in space, in regard to intensity (activity rate and duration) and amplitude. Typically the volatility/intermittency may be further classified into continuous and discrete (i.e., jumps) elements, and long and short term effects.
In finance the investigation of volatility is well developed and many of the procedures of probabilistic and statistical analysis applied are similar to those of relevance in turbulence, for instance in regard to multipower variations, particularly quadratic and bipower variations and variation ratios.
Other important issues concern the modelling of propagating stochastic volatility/intermittency fields and the question of predictability of volatility/intermittency. This paper introduces a concept of realised relative volatility/intermittency and hence of realised relative energy dissipation, the ultimate purpose of which is to assess the relative volatility/intermittency or energy dissipation in arbitrary subregions of a region C of space-time relative to the total volatility/intermittency/energy in C.
The concept of realised relative volatility/intermittency also paves the way for practical applications of some recent advances in the asymptotic theory of power variations of non-semimartingales (see, e.g., Corcuera et al. (2006) and Barndorff-Nielsen et al. (2011) to volatility/intermittency measurements and inference with empirical data. In the non-semimartingale setting, realised power variations need to be scaled properly, in a way that depends on the smoothness of the process through an unknown parameter, to ensure convergence. This makes inference, in particular, difficult. Realised relative power variations, however, are self-scaling and, moreover, admit a statistically feasible central limit theorem, which can be used, e.g., to construct confidence intervals for the realised relative volatility/intermittency. (Self-scaling statistics have also been recently used by Podolskij and Wasmuth (2013) to construct a goodness-of-fit test for the volatility coefficient of a fractional diffusion.) We start the further discussion by describing, in Section 2, how energy dissipation in turbulence is defined and traditionally assessed. This is followed by a brief outline of some results from the theory of Brownian semistationary (BSS) processes that are pertinent for the main topic of the present paper. The definition of realised relative volatility/intermittency/energy dissipation is given in Section 3. For concreteness and because of its particular importance we focus on realised relative energy dissipation. Asymptotic probabilistic properties-consistency and a functional central limit theorem-for realised relative power variations are derived in Section 4. Applications to data on turbulence and energy prices are presented in Section 5. Section 6 concludes.

Energy dissipation in turbulence
In a purely spatial setting the energy dissipation of a homogenous and isotropic turbulent field is (up to an ignorable constant involving viscosity) where y i denotes the velocity at the spatial position x ∈ R 3 . The coarse grained energy dissipation over a region C in R 3 is then given by Furthermore, if only measurements of the velocity component in the main direction x 1 of the flow are considered one defines the surrogate energy dissipation as By Taylor's frozen field hypothesis (Taylor, 1938), this may then be reinterpreted as the timewise surrogate energy dissipation which would be the relevant quantity in case the measurements were of the same, main, component of the velocity but now as a function of time rather than of position. Associated to this is the coarse grained energy dissipation corresponding to the interval [t, t + u] and given by Supposing that the velocity y t has been observed over the interval [0, T ] at times 0, δ, . . . , T /δ δ, when it comes to estimating ε + (t), as given by (2), this is traditionally done by taking the normalised realised quadratic variation Correspondingly, the coarse grained energy dissipation over [0, T ] is estimated bŷ The definitions (1) and (2) of course assume that the sample path y is differentiable. On the other hand, going back to Kolmogorov, it is broadly recognised that turbulence can only be comprehensibly understood by viewing it as a random phenomenon-see Kolmogorov (1941aKolmogorov ( ,b,c, 1962 and, for a recent overview, Tsinober (2009). Accordingly, y should be viewed as a stochastic process, henceforth denoted Y , and it is not realistic to assume that its sample paths are differentiable. Thus a broader setting for the analysis of the energy dissipation in Y is called for, and in the following we propose and discuss such a setting.
A Brownian semistationary (BSS) process, as introduced by Barndorff-Nielsen and Schmiegel (2009), may be used as a model for the timewise development of the velocity at a fixed point in space and in the main direction of the flow in a homogeneous and isotropic turbulent field. For focus and illustation we shall consider cases where Y is a stationary BSS process, where g and q are deterministic kernel functions, B is Brownian motion and σ is a stationary process expressing the volatility/intermittency of the process. In that context the gamma form of the kernel g has a special role. In particular, if ν = 5 6 and σ is square integrable, then the autocorrelation function of Y is identical to von Kármán's autocorrelation function (von Kármán, 1948) for ideal turbulence.
In relation to the BSS process (3) with gamma kernel (4) a central question is that of determining σ 2 from Y . In case the process is a semimartingale the answer is given by the quadratic variation of Y ; in fact, then is the accumulated quadratic volatility over the interval [0, t]. However, in the cases of most interest for turbulence, that is ν ∈ ( 1 2 , 1) ∪ (1, 3 2 ) the process Y is not a semimartingale and in order to determine σ 2+ by a limiting procedure from the realised quadratic variation the latter has to be normalised by a factor depending on δ and ν. Specifically, as shown by Barndorff-Nielsen and Schmiegel (2009), this factor is δc (δ) −2 where is defined using the Gaussian core Using this result for estimation of σ 2+ t requires either that ν is known or that a sufficiently accurate estimate of ν can be found, The latter question has led to detailed studies of the application of power and multipower variations to estimation of ν ( Barndorff-Nielsen et al., 2011Corcuera et al., 2013).

Realised relative V/I/E
Supposing again that the velocity Y t has been observed at times 0, δ, . . . , T /δ δ, we are interested in the relative energy dissipation of Y over any subinterval [t, t + u] where ε + (T ) is the energy dissipation in [0, T ]. Within the turbulence literature, this definition of the relative energy dissipation is strongly related to the definition of a multiplier in the cascade picture of the transport of energy from large to small scales (see Cleve et al. (2008) and references therein). We now introduce the concept of realised relative energy dissipation. Specifically, whether Y t is deterministic and differentiable or an arbitrary stochastic process we define the realised relative energy dissipation over the subinterval [t, t + u] as is the realised relative quadratic variation of Y . We note that the quantity R + δ (t, t + u) is entirely empirically based.
In the "classical" case of turbulence, where Y t is differentiable, as δ → 0 we have and hence, as δ → 0, i.e., the limit equals the relative energy dissipation (8). Now suppose that Y is a stationary BSS process (3) with gamma kernel (4) and ν > 1 2 , as it needed for the stochastic integral to exist. Then, if ν > 3 2 , Y has continuous differentiable sample paths, i.e. we are essentially in the "classical" situation. If ν = 1 the process Y is a semimartingale and the realised quadratic variation [Y δ ] converges to the quadratic variation [Y ], that is where σ 2+ t is the accumulated quadratic volatility/intermittency (5). Consequently, for the realised relative energy dissipation we have i.e., the limit is the relative accumulated squared volatility/intermittency. Finally, suppose that ν ∈ ( 1 2 , 1) ∪ (1, 3 2 ), i.e., we are in the non-semimartingale case and the sample paths are Hölder continuous of order ν − 1/2. Then, subject to a mild condition on q (see Appendix C for a result covering the case where q is of the gamma form), we have again, as δ → 0, that . This follows directly from limiting results of Barndorff-Nielsen and Schmiegel (2009) and Barndorff-Nielsen et al. (2011). In view of these results, in the turbulence context we view the limit of R + δ as the relative energy dissipation. Remark 1. As mentioned in Section 2, use of the original assessment procedure (7) requires determination of the degree of freedom/smoothness parameter ν. The realised relative quadratic variation [Y δ ] t,T is entirely empirically determined, self-scaling, and its consistency does not rely on inference on ν.

Asymptotic theory of realised relative power variations
We develop now a probabilistic asymptotic theory for realised relative power variations, going slightly beyond the earlier discussion of quadratic variations and energy dissipation. To highlight the robustness of realised relative power variations to model misspecification, we consider both a BSS process and a Brownian semimartingale as the underlying process. While we limit the discussion to power variations for the sake of simpler exposition, our results can be easily extended to multipower variations.

Probabilistic setup and consistency
Let us consider a stochastic process Y = {Y t } t≥0 , defined on a complete filtered probability space (Ω, F, (F t ) t∈R , P ) via the decomposition where A = {A t } t≥0 is a process that allows for skewness in the distribution of Y t . The process A is assumed to fulfill one of two negligibility conditions, viz. (10) and (13) given below (Appendix C presents more concrete criteria that can be used to check these conditions). Given a standard Brownian motion B = {B t } t∈R and a càglàd process σ = {σ t } t∈R , adapted to the natural filtration of B, we allow for the following two specifications of the process X = {X t } t≥0 .
(I) X is a local Brownian martingale given by (II) X is a BSS process given by for all t ≥ 0.
On the one hand, by choosing A to be absolutely continuous in the case (I), we see that this framework includes rather general Brownian semimartingales. On the other hand, in the case (II) the process Y is typically a non-semimartingale, as discussed above.
Remark 2. In addition to the setting (II), asymptotic theory for relative realised power variations could also be developed in the context of the fractional processes studied by Corcuera et al. (2006). Then, we would define where {Z H t } t≥0 is a fractional Brownian motion with Hurst parameter H ∈ 1 2 , 3 4 and σ satisfies certain path-regularity conditions (Corcuera et al., 2006, pp. 716, 723). The proofs would be analogous to the case (II), but take as an input the asymptotic results of Corcuera et al. (2006), instead of those of Barndorff-Nielsen and Schmiegel (2009) and Barndorff-Nielsen et al. (2011).
Recall that for any p > 0, the p-th order realised power variation of the process Y with lag δ > 0 is given by The power variations [A δ ] (p) and [X δ ] (p) are, of course, defined analogously. Similarly to earlier literature on power and multipower variations of BSS processes ( Barndorff-Nielsen et al., 2011Corcuera et al., 2013) we assume that the kernel function g behaves like t ν−1 near zero for some ν ∈ ( 1 2 , 1) ∪ (1, 3 2 ), or more precisely that where L g is slowly varying at zero, which implies that X is not a semimartingale in the case (II). Then, under some further regularity conditions on g (Corcuera et al., , pp. 2555(Corcuera et al., -2556, which include the assumption that c(δ) = δ ν−1/2 L c (δ) with L c slowly varying at zero, and which are satisfied for instance when g is the gamma kernel (4), we have where c(δ) is given by (6), σ p+ t = t 0 |σ s | p ds, and m p = E{|ξ| p } for ξ ∼ N (0, 1), by Theorem 3.1 of Corcuera et al. (2013). In the case (I), setting c(δ) = √ δ, the convergence (9) holds without any additional assumptions, e.g., by Theorem 2.2 of Barndorff-Nielsen et al. (2006). Additionally, note that the convergence (9) holds also when X is replaced For fixed time horizon T > 0, we introduce the p-th order realised relative power variation process over [0, T ] by The relative power variation has the following evident consistency property.

Central limit theorem
Recall first that random elements U 1 , U 2 , . . . in some metric space U converge stably (in law) to a random element U in U, defined on an extension (Ω , F , P ) of the underlying probability space (Ω, F, P ), if for any bounded, continuous function f : U → R and bounded random variable V on (Ω, F, P ). We denote stable convergence by st − →.
Remark 6. Stable convergence, introduced by Rényi (1963), is stronger than ordinary convergence in law and weaker than convergence in probability. It is key that the limiting random element U is defined on an extension of the original probability space, because in the case where U is F-measurable, the convergence U n st − → U is in fact equivalent to U n p − → U (Podolskij and Vetter, 2010, Lemma 1).
Remark 7. The usefulness of stable convergence can be illustrated by the following example that is pertinent to the asymptotic results below. Suppose that U n st − → θξ in R, where ξ ∼ N (0, 1) and θ is a positive random variable independent of ξ. In other words, U n follows asymptotically a mixed Gaussian law with mean zero and conditional variance θ 2 . Ifθ n is a positive, consistent estimator of θ, i.e.,θ n p → θ, then the stable convergence of U n allows us to deduce that U n /θ n d − → N (0, 1). We refer to Rényi (1963), Aldous and Eagleson (1978), Jacod and Shiryaev (2003, pp. 512-518), and Podolskij and Vetter (2010, pp. 332-334) for more information on stable convergence.
Let us write D([0, T ]) for the space of càdlàg functions from [0, T ] to R, endowed with the usual Skorohod metric (Jacod and Shiryaev, 2003, Chapter V). (Recall, however, that convergence to a continuous function in this metric is equivalent to uniform convergence.) Under slightly strengthened assumptions, the realised power variation of X satisfies a stable central limit theorem of the form where λ X,p > 0 is a deterministic constant and {W t } t∈[0,T ] a standard Brownian motion, independent of F, defined on an extension of (Ω, F, P ). Indeed, in the case (I) the convergence (11) holds with λ X,p = m 2p − m 2 p , provided that σ is an Itô semimartingale ( Barndorff-Nielsen et al., 2006, Theorem 2.4). Moreover, we have (11) also in the case (II) if we make the restriction ν ∈ ( 1 2 , 1), the situation of most interest concerning turbulence, and assume that σ satisfies a Hölder condition in expectation (Corcuera et al., 2013, Theorem 3.2). Then, in contrast to the semimartingale case, where λ p : ( 1 2 , 1) → (0, ∞) is a continuous function defined using the correlation structure of fractional Brownian noise (see Appendix B for the definition and proof of continuity). Analogously to (10), the convergence (11) The realised relative power variation of Y satisfies the following central limit theorem, which is an immediate consequence of Lemma 14 in Appendix A.
Remark 9. In the case (II) the restriction ν ∈ ( 1 2 , 1) can be relaxed when one considers power variations defined using second or higher order differences of Y ( Corcuera et al., 2013). Then, (11) holds for all ν ∈ ( 1 2 , 1)∪(1, 3 2 ). As Theorems 3 and 8 do not depend on the type of differences used in the power variation, they obviously apply also in this case.
Conditional on F, the limiting process on the right-hand side of (14) is a Gaussian bridge. In particular, its (unconditional) marginal law at time t ∈ [0, T ] is mixed Gaussian with mean zero and conditional variance Note also that when σ is constant, the limiting process reduces to a Brownian bridge. In effect, the result is analogous to Donsker's theorem for empirical cumulative distribution functions (see, e.g., Kosorok (2008) for an overview of such results). Clearly, we may estimate the asymptotic variance (15) consistently using In the case (II) the estimator V t (δ) is not feasible as such since ν appears as a nuisance parameter in λ X,p = λ p (ν). However, we may replace λ X,p with λ p (ν δ ), whereν δ is any consistent estimator of ν based on the observations Y 0 , Y δ , . . . , Y T /δ δ (they have been developed by Barndorff-Nielsen et al. (2011; Corcuera et al. (2013)). Using the properties of stable convergence, we obtain the following feasible central limit theorem.

Inference on realised relative V/I/E
Proposition 10 can be used to construct approximative, pointwise confidence intervals for the relative volatility/intermittency σ p+ t,T . Since, by construction, σ p+ t,T assumes values in [0, 1], it is reasonable to constrain the confidence interval to be a subset of [0, 1]. Thus, we define for any a ∈ (0, 1) the corresponding (1 − a) · 100% confidence interval as where z 1−a/2 > 0 is the 1 − a 2 -quantile of the standard Gaussian distribution. Another application of the central limit theory is a non-parametric homoskedasticity test that is similar in nature to the classical Kolmogorov-Smirnov and Cramér-von Mises goodness-of-fit tests for empirical distribution functions. This extends the homoskedasticity tests proposed by Dette et al. (2006) and Dette and Podolskij (2008) to a nonsemimartingale setting. Another extension of these tests to non-semimartingales, namely fractional diffusions, is given by Podolskij and Wasmuth (2013). The approach is also similar to the cumulative sum of squares test (Brown et al., 1975) of structural breaks studied in time series analysis. To formulate our test, we introduce the hypotheses H 0 : σ t = σ 0 for all t ∈ [0, T ], H 1 : σ t = σ 0 for some t ∈ [0, T ].
As mentioned above, Theorem 8 implies that under H 0 , The distance between the realised relative power variation and the linear function can be measured using various norms and metrics. Here, we consider the typical sup and L 2 norms that correspond to the Kolmogorov-Smirnov and Cramér-von Mises test statistics, respectively. More precisely, we define the statistics Note that also in (17), we may use λ p (ν δ ) instead of λ X,p in the case (II). By (16) and the scaling properties of Brownian motion, we obtain under H 0 the classical Kolmogorov-Smirnov and Cramér-von Mises limiting distributions for our statistics, namely, Remark 11. Well-known series expansions for the cumulative distribution functions of the limiting functionals in (18) can be found, e.g., in Lehmann and Romano (2005, p. 585) and Anderson and Darling (1952, p. 202).
Remark 12. The finite-sample performance of the test statistics S KS δ and S CvM δ is explored in a separate paper (Bennedsen et al., 2014).

Brookhaven turbulence data
We apply the methodology developed above first to data of turbulence. The data consist of a time series of the main component of a turbulent velocity vector, measured at a fixed position in the atmospheric boundary layer using a hotwire anemometer, during an approximately 66 minutes long observation period at sampling frequency of 5 kHz (i.e. 5000 observations per second). The measurements were made at Brookhaven National Laboratory (Long Island, NY), and a comprehensive account of the data has been given by Drhuva (2000).
As a first illustration, we study the observations up to time horizon T = 800 milliseconds. Using the smallest possible lag, δ = 0.2 ms, this amounts to 4000 observations. Figure 1(a) displays the squared increments corresponding to these observations. As a comparison, the same time horizon is captured in Figure 1(b) but with lag δ = 0.8 ms. Figure 1(c) compares the associated accumulated realised relative energy dissipations/quadratic variations. The graphs for these two lags show very similar behaviour, exhibiting how the total time interval is divided into a sequence of intervals over which the slope of the energy dissipation is roughly constant. On the other hand, the amplitudes of the volatility/intermittency are of the same order in the whole observation interval.
To be able to draw inference on relative volatility/intermittency using the data, we need to address two issues. Firstly, for this time series, the lags δ = 0.2 ms and δ = 0.8 ms are below the so-called inertial range of turbulence, where a BSS process with a gamma kernel, a model of ideal turbulence, provides an accurate description of the data-see Corcuera et al. (2013), where the same data are analysed. Secondly, the data were digitised using a 12-bit analog-to-digital converter. Thus, the measurements can assume at most 2 12 = 4096 different values, and due to the resulting discretisation error, a non-negligible amount of the increments are in fact equal to zero (roughly 20 % of all increments). These  discretisation errors are bound to bias the estimation of the parameter ν, which is needed for the inference methods. We mitigate these issues by subsampling, namely, we apply the inference methods using a considerably longer lag, δ = 80 ms, which is near the lower bound of the inertial range for this time series   Figure 1).
We divide the time series into 66 non-overlapping one-minute-long subperiods, testing the constancy of σ, i.e., the null hypothesis H 0 , within each subperiod. Figure 2(a) displays the estimates of ν for each subperiod using the change-of-frequency method ( Corcuera et al., 2013). All of the estimates belong to the interval ( 1 2 , 1) and they are scattered around the value ν = 5 6 predicted by Kolmogorov's (K41) scaling law of turbulence (Kolmogorov, 1941a,b). The homoskedasticity test statistics, for p = 2, and their critical values, derived using (18), in Figure 2 To understand what kind of intermittency the tests are detecting in the data, we look into two extremal cases, the 27th and 40th subperiods (the red bars in Figure 2(b) and (c)). To this end, we plot the realised relative energy dissipations, with δ = 80 ms, during the 27th and 40th subperiods in Figure 3(a) and (b), respectively. We also include the pointwise confidence intervals, the p-values of the homoskedasticity tests, and as a reference, the realised relative energy dissipations using the smallest possible lag δ = 0.2 ms. While the realised relative energy dissipations exhibit a slight discrepancy between the lags δ = 80 ms and δ = 0.2 ms, it is clear that 40th subperiod indeed contains significant intermittency, whereas the during the 27th subperiod, the (accumulated) realised relative energy dissipation grows nearly linearly.

EEX electricity spot prices
We also briefly exemplify the concept of relative volatility using electricity spot price data from the European Energy Exchange (EEX). Specifically, we consider deseasonalised daily Phelix peak load data (that is, the daily averages of the hourly spot prices of electricity delivered between 8 am and 8 pm) with delivery days ranging from January 1, 2002 to October 21, 2008. Weekends are not included in the peak load data, and in total we have 1775 observations. This time series was studied in the paper by  and the deseasonalisation method is explained therein. As usual, we consider here logarithmic prices. Figure 1(d) shows the squared increments up to the total time horizon T = 1775 days with lag δ = 1 day. The same time horizon is captured in Figure 1(e) but with a resolution δ = 4 days. Figure 1(f) compares the corresponding accumulated realised relative quadratic variations. The results for these two lags do not show the same similarity as with the turbulence data (Figure 1(a-b)). Judging by eye, we observe that the intensity of the volatility is changing with lag δ. This lag dependence is also observed in the amplitudes, again in contrast to the figures on the left hand side. (However, more quantitative investigation of such amplitude/density arguments is outside the scope of the present paper.) The dependence of the estimation results on the lag δ is, at least partly, explained by the relatively low sampling frequency of the data. With δ = 1 day, the increments are dominated by a few exceptional observations (which may correspond  Remark 13. It was shown by  that by suitably choosing both g and q to be of gamma type it is possible to construct a BSS process with normal inverse Gaussian one-dimensional marginal law, which corresponds closely to the empirics for the time series of log spot prices considered. Moreover, the estimated value of the smoothness parameter ν for this time series falls in the interval ( 1 2 , 1).

Conclusion
The definition of realised relative energy dissipation introduced in this paper applies to any continuous time, real valued process Y . An extension to vector valued processes is an  Figure 3: Brookhaven turbulence data: Realised relative energy dissipation during the 27th (a) and 40th (b) subperiods with δ = 80 ms and δ = 0.2 ms. Additionally, p-values for the hypothesis H 0 , estimates of ν using the change-of-frequency method, and 95% pointwise confidence intervals, all using the lag δ = 80 ms.
issue of interest, in particular in relation to the definition (1) of the energy dissipation in three-dimensional turbulent fields.
The extent to which the realised volatility/intermittency/energy is an empirical counterpart of what can be conceived theoretically as relative volatility/intermittency/energy depends on the model under consideration. As discussed above this is the case, in particular, both under Brownian semimartingales, as these occur widely in mathematical finance and financial econometrics, and under stationary BSS processes.
In the timewise stationary setting, the realised relative energy dissipation is a parameter free statistic which provides estimates of the relative energy in subintervals of the full observation range, by relating the quadratic variation over each subinterval to the total realised energy for the entire range. It provides robust estimates of the relative accumulated energy as this develops over time and is intimately connected to the concepts of volatility/intermittency and energy dissipation as these occur in statistical turbulence and in finance. This was illustrated in connection to the class of BSS processes with g of the gamma form.
Lemma 15. The function λ p is continuous.

C Sufficient conditions for the negligibility of the skewness term
Suppose first that the process A = {A t } t≥0 is given by where µ ∈ R is a constant and the process {a t } t≥0 is measurable and locally bounded. Then we can establish rather simple conditions for its negligibility in the asymptotic results for power variations. By Jensen's inequality, we have for any p ≥ 1, s ≥ 0, and t ≥ 0, where C a > 0 is a random variable that depends locally on the path of a. Thus, the condition (10) holds whenever δ c(δ) → 0, and (13) holds if δ p−1/2 c(δ) p → 0.
Suppose now, instead, that A follows where q is the gamma kernel q(t) = c t η−1 e −ρt for some c > 0, η > 0, and ρ > 0. We assume that the process {a t } t∈R is measurable, locally bounded, and satisfies for any t ≥ 0, which is true, e.g., when the auxiliary process u −∞ q(u − s)|a s |ds, u ≥ 0, has a càdlàg or continuous modification.
Next, we want to show that In the case η ≥ 2 the derivative q is bounded and (28) is immediate. Suppose that η < 2. Then, |q (t)| ≤ Ct η−2 on any finite interval, where C > 0 depends on the interval. Using the mean value theorem, we obtain which implies (28). To bound |I 4 δ |, note that, by (27), |q (t)| ≤ C q(t) for all t ≥ −s * , where C > 0 is a constant. For any s < s * , we have (j − 1)δ − s > η−1 ρ . Thus, by the mean value theorem, q(jδ − s) − q((j − 1)δ − s) ≤ C q (j − 1)δ − s δ and, consequently, Collecting the estimates, we have Checking the sufficiency of the asserted conditions is now a straightforward task.