Overfitting of Hurst estimators for multifractional Brownian motion: A fitting test advocating simple models

Abstract

The aim of this paper is to provide simple models with a time-varying Hurst index. Such models should be simple as much as possible and well fit the estimated Hurst index. After a recall on the fractional and multifractional Brownian motion and on the statistical estimation of the Hurst index, we propose a fitting test for a model with a time-varying Hurst index. Then, we give an approach to select a simple model. Our approach is illustrated by numerous numerical simulations and then applied to market finance data.

Keywords

Fractional Brownian motion (fBm)multifractional Brownian motion (mBm)time-varying Hurst index wavelet series expansion finance time series

1. Introduction

As the data become larger, the stochastic models to describe datasets become more and more elaborated. This paper concerns large or big time series, modeled by multifractional Brownian motion (mBm), mainly with applications to financial series. However, our results can be applied to other practical fields. Recall that mBm are fractional Brownian motion (fBm) where the constant Hurst index has been replaced by a time-varying Hurst index. Our claim is that mBm with a Hurst index being itself a stochastic process is an useless generalization. Statistics, more precisely sampling fluctuation, explain that, even for infinitely differentiable Hurst index, the usual time-varying Hurst index estimators are stochastic processes. Moreover, smooth enough time-varying Hurst index is relevant for many applications, whereas the more general model looks like a theoretical challenge for smart probabilist. So, we advocate more simple models with infinitely differentiable or piecewise smooth Hurst index.

The rest of this article is organized as follows. In Section 2, we give recalls on the fractional Brownian motion and multifractional Brownian motion. Then we give some statistics for fBm and mBm. After that, we explain the fitting test for a time-varying Hurst index in Section 3. In Section 4, we estimate the Hurst index for fBm and mBm and we give some results of numerical simulations using a software improved based on the one of [7]. Finally, in Section 5, we apply our approach on two real financial datasets.

2. Fractional and multifractional Brownian motion

The Brownian motion was introduced by Bachelier (1901) for stock options in finance and next by Einstein (1905) in order to describe successive movements of atomic particles independent one from another. The mathematical theory was developed by Wiener during the 1920’s. In 1940, Kolmogorov has introduced the fractional Brownian motion, which was popularized by Mandelbrot since the 1960’s [35], the precise definition is postponed in Section 2.1. At these times (during 1970’s and 1980’s), we only have access to small time series. For example, the CAC 40 (Cotation Assistée en Continu = Electronic Continuous time trading) was adopted at Paris Bourse in 1987. Similar technological progresses arise in other practical fields. Time series observed at higher frequencies point that Hurst index is not constant but often vary over time. The multifractional Brownian motion was then introduced in [10,39]. The mBm is a fBm where the Hurst index H becomes a time-varying function $H (t)$ , but must be smooth enough and satisfies a technical condition. The introduction of the multifractional Brownian motion has then allowed the application in several fields like finance [13–15], turbulence [38], magnetospheric dynamics [43], biology [36].

2.1. Recall on fractional Brownian motion

In this paper, we aim to give a method for selecting a good probabilistic model with a time-varying Hurst index. To begin, we give an overview of the process we are working with. The multifractional Brownian motion (mBm) can be viewed as a generalisation of the fractional Brownian motion (fBm), which is defined as follows:

Definition 2.1.
The fBm ${B_{H} (t), t \in R}$ with Hurst index $H \in (0, 1]$ and scale factor σ is the zero mean Gaussian process which covariance function is $\begin{matrix} (2.1) & \begin{matrix} cov [B_{H} (s), B_{H} (t)] \\ = \frac{σ^{2}}{2} {| s |^{2 H} + | t |^{2 H} - | t - s |^{2 H}} \\ for all (s, t) \in R^{2} . \end{matrix} \end{matrix}$

The fBm has stationary increments, it admits different representations, see e.g. [12], and can be also viewed as a Gaussian field depending both on the time t and the Hurst index H. For instance, the harmonisable representation of the fBm considered as a Gaussian field is given by $\begin{matrix} (2.2) & \begin{matrix} B (t, H) = \int_{R} \frac{(e^{i t ξ} - 1)}{| ξ |^{H + 1 / 2}} d W (ξ) \\ for all t \in R, \end{matrix} \end{matrix}$ where $(t, H) \mapsto B (t, H) : = B_{H} (t)$ is the fBm, and W is a complex valued Wiener measure such that $B (t, H)$ is real valued. The Hurst index H drives three properties: pathwise regularity, self-similarity, correlation of the increments, see e.g. [12]. Figure 1 below shows a simulated path of fractional Brownian motion with constant Hurst index for $H = 0.75$ .
2.2. Recall on multifractional Brownian motion

The mBm is a generalisation of the fBm where the Hurst index H is replaced by a time-varying function $H (t)$ , as stated by the following definition [5]:

Definition 2.2.
Let $(t, H) \mapsto B (t, H)$ be the Gaussian field defined by (2.2). The multi-fractional Brownian motion can be defined by $\begin{array}{rcl} (2.3) & X (t) & = & B (t, H (t)) for all t \in R . \end{array}$

Due to the time-varying Hurst index, stationarity of the increments does not hold anymore, therefore long range dependency becomes a meaningless notion. Similarly, global self-similarity no more holds, but mBm verifies local asymptotic self-similarity of order $H (t)$ . In fact, mBm is tangent at time t to a fBm of Hurst index $H (t)$ . For the mBm, the roughness of the paths corresponds to the Hurst index $H (t)$ . However, this property is satisfied under an extra-condition insuring that the Hölder regularity of the time-varying Hurst index $t \mapsto H (t)$ is greater than the maximum value of $H (t)$ [5,10,12]. In a nutshell, the Gaussian field $(t, H) \mapsto B (t, H)$ is $C^{H}$ with respect to the first variable t, but $C^{\infty}$ with respect to the second variable H. Thus the mBm regularity on any open interval is the minimum of $H (t)$ and the regularity of the map $t \mapsto H (t)$ .

Figures 1 and 2 below show a path of a multifractional Brownian motion with two Hurst index functions. The software used is based on a code of [6] that we have improved for our study.

Fig. 1.
A path of mBm with Hurst index $H (t) = 0.75$ .

Fig. 2.
A path of mBm with Hurst index $H (t) = 0.5 + 0.4 \times sin (5 t)$ .

Fig. 3.
Mean estimation of constant function Hurst index $H (t) = 0.75$ with $α = 0.35$ .

Fig. 4.
Mean estimation of Hurst index function $H (t) = 0.25 + 0.45 \times {sin}^{2} (t)$ with $α = 0.35$ .
2.3. Statistics of fBm and mBm

Let X be a fBm or a mBm. We observe one path of size n of the process X with mesh $h_{n} = \frac{1}{n}$ , namely $(X (t_{k}))$ with $t_{k} = k / n$ , for $k = 1, \dots, n$ . We want to estimate the Hurst index H, respectively the time-varying Hurst index $H (t)$ . There are many methods to estimate the Hurst index for a fBm. Let us mention detrended fluctuation analysis (DFA), Wavelet coefficients ${\hat{H}}^{(W)}$ [1], Linear Regression of Generalized Quadratic Variation ${\hat{H}}^{(GQV)}$ [9,17], or Increment Ratio Statistics ${\hat{H}}^{(IR)}$ [6,11]. The standard method for estimating a time-varying Hurst index for mBm localises the estimation of a constant Hurst index on a small vicinity of each time t, namely on $\begin{matrix} V (t, ε_{n}) = {t_{k} such that | t_{k} - t | ⩽ ε_{n}}, \end{matrix}$ where $ε_{n} = n^{- α}$ , with $0 < α < 1$ .

It has been pointed in [12,24] that the localisation procedure of Hurst estimator adds a stochastic process to the right Hurst index $H (t)$ . Figures 3 and 4 show in blue the function of the Hurst index, in green the estimation of the mean Quadratic Variation and in red the estimation of the mean Increment Ratio Statistics.

For Linear Regression of GQV and Increment Ratio Statistics, this sampling fluctuation has been proved through functional Central Limit Theorems (CLT) in [7,17,18].

Theorem 2.3 (CLT Coeurjolly, 2005–2006 [17,18]).

Let X be a mBm with Hurst index $0 < H (t) < 1$ where $H \in C^{1} ([0, 1], (0, 1))$ ; then for all time t, $H_{n}^{(GQV)} (t) ⟶ H (t)$ almost surely, and $\begin{matrix} (2.4) & \begin{matrix} \sqrt{2 ε_{n} \cdot n} \times ({\hat{H}}_{n}^{(GQV)} (t) - H (t)) \\ \overset{[}{} n \to \infty] D ⟶ G^{'} (t), \end{matrix} \end{matrix}$ where $G^{'} (t)$ a zero mean Gaussian process, with covariance structure: $\begin{array}{l} (2.5) & \begin{matrix} Var (G^{'} (t)) = γ (H (t)) \\ for all t \in (0, 1), \end{matrix} \\ (2.6) & \begin{matrix} cov (G^{'} (t_{1}), G^{'} (t_{2})) = 0 \\ with (t_{1}, t_{2}) \in {(0, 1)}^{2} for t_{1} \neq t_{2} . \end{matrix} \end{array}$

Theorem 2.4 (CLT Bardet–Surgailis 2013 [7]).

Under some technical assumptions the functional CLT of Coeurjolly [ 17 ] holds for both GQV and IR estimator, with two different limit processes $G^{(GQV)} (t)$ and $G^{(IR)} (t)$ . For all family $t_{1}, \dots, t_{k}$ , $G^{(IR)} (t_{i})$ are independent centered Gaussian random variables such that $\begin{matrix} E {[G^{(IR)} (t_{i})]}^{2} = γ^{(IR)} (H (t)) \end{matrix}$ and $G^{(GQV)} (t_{i})$ are independent centered Gaussian random variables such that $\begin{matrix} E {[G^{(GQV)} (t_{i})]}^{2} = γ^{(GQV)} (H (t)) . \end{matrix}$

As a consequence of the functional CLT in [7,17], we can quote:

1.
$E {\hat{H}}_{n}^{(GQV)} (t) \to H (t)$ , since $G^{(GQV)} (t)$ is a zero mean process.
2.
$E {\hat{H}}_{n}^{(IR)} (t) \to H (t)$ , since $G^{(IR)} (t)$ is a zero mean process.
3.
The speed of convergence is $\sqrt{2 ε_{n} \cdot n}$ , which allows us to deduce confidence intervals.
The CLT’s in [7 ,17] explain the sampling fluctuation of the time-varying Hurst estimators for GQV and IRS, but we guess that the same kind of phenomenon remains in force for other estimators.
3. A fitting test for time-varying Hurst index of mBm

When we estimate a time-varying Hurst index $H (t)$ by localisation of an estimator of constant Hurst index, we observe a sampling fluctuation of the estimate $\hat{H} (t)$ around the right Hurst index $H (t)$ . The localised estimator $\hat{H} (t)$ overfits the data by learning both the right Hurst index and also the sampling fluctuation. In this section, we propose a fitting test for a time-varying Hurst index and apply it to a model selection approach.

3.1. Fitting test

From the functional CLT (Theorem 2.4), we deduce the convergence of normalized square error as stated in the following Central Limit Theorem:

Proposition 3.1.
Under the assumptions of Theorem 2.4 , we have $\begin{array}{l} {(\frac{1}{L}) \times \sum_{ℓ = 1}^{L} [(2 n ε_{n}) \times {| {\hat{H}}_{n}^{(GQV)} (t_{ℓ}) - H (t_{ℓ}) |}^{2} \\ - γ^{(GQV)} (H (t_{ℓ}))]} \\ / {[(\frac{1}{L}) \times \sum_{ℓ = 1}^{L} γ^{(GQV)} {(H (t_{ℓ}))}^{2}]}^{1 / 2} \\ \overset{[}{} L \to \infty, n \to \infty] D ⟶ N (0, 1), \end{array}$ 1.
The convergence in distribution with respect to the two parameters L and n means that the cumulative distribution function of the left hand is approximatively (up to ε) the one of standard Gaussian law, for L large enough and $n > N_{0} (L)$ .
2.
Moreover, $H (t)$ is the right Hurst index,
3.
and $γ^{(GQV)} (H)$ is given by the functional CLT for ${\hat{H}}^{(GQV)} (t)$ .

By using Proposition 3.1, we can test if a time-varying Hurst index $\tilde{H} (\cdot)$ is an admissible model.
Theorem 3.2.
Under the same assumptions and with the same notations as in Proposition 3.1 , we can test $\begin{array}{l} (H_{0}) : \tilde{H} (\cdot) = H (\cdot) versus \\ (H_{1}) : \tilde{H} (\cdot) \neq H (\cdot) . \end{array}$ Under the null assumption, $T_{n} (\tilde{H} (t)) \overset{[}{} L \to \infty, n \to \infty] D ⟶ N (0, 1)$ , where $T_{n} (\tilde{H} (t))$ is defined by $\begin{matrix} T_{n} (\tilde{H}) = & {(\frac{1}{L}) \times \sum_{ℓ = 1}^{L} [(2 n ε_{n}) \\ \times {| {\hat{H}}_{n}^{(GQV)} (t_{ℓ}) - \tilde{H} (t_{ℓ}) |}^{2} \\ - γ^{(GQV)} (\tilde{H} (t_{ℓ}))]} \\ / {[(\frac{1}{L}) \times \sum_{ℓ = 1}^{L} γ^{(GQV)} {(\tilde{H} (t_{ℓ}))}^{2}]}^{1 / 2} . \end{matrix}$
Proof.
Theorem 3.2 is a direct consequence of Proposition 3.1. The numerical simulations in Section 4 confirm the theorem. We give a sketch of the proof in the Appendix. □

Let us stress that we cannot calculate the power of the test since $H (\cdot) \in C ([0, 1])$ which is an infinite dimensional vector space.
3.2. Application to model selection

The naive time-varying estimator $\hat{H} (t)$ is the sum of the right Hurst index $H (t)$ and a Gaussian process with irregular trajectories. A natural solution consists in smoothing the naive estimator. We can use a kind of Portemanteau test to select an admissible model. For instance:

$M_{0}$ the family of constant models $\tilde{H} (t) = H$ .

$M_{1}$ the family of affine models $\tilde{H} (t)$ .

$M_{2}$ the family of quadratic models $\tilde{H} (t)$ .

$M_{3}$ the family of piecewise quadratic models $\tilde{H} (t)$ .

We successively test models extracted from the previous families. Those families of models are classified by order of complexity of function

\tilde{H} (t)

. We stop and use the first eligible model, namely for family

M_{i}

with the lowest i. By construction of these families, the selected model is thus the simplest one.

4. Numerical simulations

A simulation must meet the following specifications: for a benchmark of time-varying Hurst indices $H (t)$ , we can simulate M paths of mBm $X (t) = B (t, H (t))$ and the different estimators $\tilde{H} (t)$ should be close enough of the right Hurst index $H (t)$ , at least the empirical mean.

We use the benchmark corresponding to [6,7,29] $H (t) = 0.75$ , $H (t) = 0.1 + 0.8 \times t$ , $H (t) = 0.5 + 0.4 \times sin (5 t)$ , $H (t) = 0.25 + 0.45 \times [1 - t \times {(sin (10 t))}^{2}]$ .

It is comfortable, but not essential, to have a small computing time. We have tested simulations by using Fractlab 2.2 and the code written by Jean-Marc Bardet [6,7] that we have improved for our study.

We test GQ estimator, decimated IRS2 estimators as in Bardet and Surgailis, (non decimated) IRS1 estimator and polynomial interpolation of (non decimated) IRS1.

Fig. 5.

One path of the estimators for Hurst index $H (t) = 0.75$ .

We note that the polynomial interpolation of the IRS estimator, for a single simulation (Fig. 5) or on average (Fig. 6), is the closest estimator to the right value of the Hurst index function. The IRS estimator alone is also a good approximation for the Hurst index despite fluctuations. The decimated IRS of Bardet [7] has a strong fluctuation that is larger than the Quadratic Variation estimator. Moreover, the curve of the polynomial interpolation of the IRS in yellow is the one that fluctuates the least compared to the other estimators.

Fig. 6.

Average estimators for Hurst index $H (t) = 0.75$ .

Next, we consider an affine function $H (t) = 0.1 + 0.8 t$ . We remark that the best estimate of the right value of the Hurst index is again the polynomial interpolation of the IRS estimator in yellow for a single simulation (Fig. 7) but also for the average of the Monte Carlo simulations (Fig. 8).

Fig. 7.

One path of the estimators for Hurst index $H (t) = 0.1 + 0.8 t$ .

Fig. 8.

Average estimators for Hurst index $H (t) = 0.1 + 0.8 t$ .

Eventually, we test on two sinusoidal functions $H (t) = 0.5 + 0.4 \times sin (5 t)$ (Fig. 9 and Fig. 10) and $H (t) = 0.25 + 0.45 \times [1 - t \times {(sin (10 t))}^{2}]$ (Fig. 11 and Fig. 12).

Fig. 9.

One path of the estimators for Hurst index $H (t) = 0.5 + 0.4 \times sin (5 t)$ .

Fig. 10.

Average estimators for Hurst index $H (t) = 0.5 + 0.4 \times sin (5 t)$ .

In the case of a single simulation, we notice that the polynomial interpolation of the IRS is the best estimator to approximate the right value of the Hurst index function $H (t)$ . However, we can also consider that the Bardet HQV estimator and Surgailis [7] and the IRS estimator are good despite their fluctuations. By averaging the simulations, we notice that we can consider the decayed IRS estimator for the right Hurst index function as good despite the fluctuations we see in a single simulation.

Then we calculate the bias and the Integrated Square Error (ISE) for each function, see e.g. Fig. 13 when $H (t) = 0.75$ .

Fig. 11.

One path of the estimators for Hurst index $H (t) = 0.25 + 0.45 \times [1 - t \times {(sin (10 t))}^{2}]$ .

We remark that:

Fig. 12.

Average estimators for Hurst index $H (t) = 0.25 + 0.45 \times [1 - t \times {(sin (10 t))}^{2}]$ .

Fig. 13.

ISE vs Bias for different Hurst index estimators, when $H (t) = 0.75$ .

Fig. 14.

ISE vs Bias for different Hurst index estimators, when $H (t) = 0.1 + 0.8 t$ .

Fig. 15.

Average bias vs ISE or the different Hurst index estimators, when $H (t) = 0.5 + 0.4 \times sin (5 t)$ .

When the Hurst index $H (t)$ is smooth, polynomial interpolation gives smaller MISE (Mean Integrated Square Error) than IRS or GQV estimators, see Fig. 14 and Fig. 15.

Even when $H (t)$ is rough, for example when $H (t)$ is itself a fBm, the polynomial interpolation and the naive IRS have the same MISE.

All estimators always give wrong estimation of the regularity (or roughness) of $H (t)$ .

If we guess the good degree of interpolation, we would greatly reduce MISE.

5. Applications

5.1. Some applications outside finance

The multifractional Brownian motion (mBm) is hence defined from the fBm but with a time-varying Hurst index, which can be encountered in many different kinds of applications:

In turbulence, Papanicolaou and Sølna [38] denote that “the power law itself [i.e. the Hurst index, …] and the multiplicative constant are not constants but vary slowly” in [38], whereas Lee [31] uses mBm with a regularly time-varying Hurst index for the air velocity, see [31, Fig. 5, p. 103].

Statistical study on magnetospheric dynamics points that an abrupt change in Hurst index can be observed a few hours before a space storm in solar wind [43].

In systems biology, [36] uses mBm to simulate molecular crowding that matches the statistical properties of sample data, whereas [33] uses mBm with piecewise constant Hurst index to model single file diffusion that is the motion of chemical, physical or biological particles in quasi-one-dimensional channel.

5.2. Applications in finance

Quantitative finance has been the most important field of application of both time series and stochastic processes for the last fifty years. The modern theory of modern finance is based on a fundamental concept defined by Fama [21]: the totality of available information is contained in market prices. New information will be immediately incorporated in the current prices. Fama distinguishes different form of market efficiency [22]. The informational efficiency could be strong when the set of information includes all the public and private information. A less restrictive notion is the semi-strong efficiency when the information set excludes the private information. But, the empirical tests aim to test a weak form of efficiency where the available information set includes only the past variations of the prices. The null hypothesis is hence the unpredictability of the future prices from the past prices [30]. The informational efficiency implies the impossibility from an agent to perform systematically better than the market. It is worth noting that this concept results from the combination of two economic hypotheses: (a) the rational expectation hypothesis according to which the agents use optimally the information and do not make systematic error [37] and (b) the market equilibrium condition: the observed price reflects the fundamental value of the asset. This last hypothesis could result from the great numbers of agents on the market (atomicity hypothesis). In other words, a market is efficient in the weakly form, if it is not possible to make (above average) profits (without accepting above average risk) by trading on the base of the past prices information set [34]. From an empirical point of view, the random walk hypothesis or the less restrictive martingale hypothesis is tested [28]. We refer to [32] for a survey of the different econometric methods.

The efficient market hypothesis [22] has lead to reject fBm as an admissible model for stock price. So, since the 1970’s, the use of martingale models with constant Hurst index $H = 1 / 2$ has become mainstream in quantitative finance [20].

The main objection to fBm, as an admissible model for stock prices, is the existence of an arbitrage opportunity for such a fBm with a constant and known Hurst index. To put it in a nutshell, an arbitrage opportunity means the possibility of producing a positive return from zero investment by clever trading. For a fBm with known and constant Hurst index, it is possible to make an arbitrage, with a strategy based on infinitely small meshes of times and without transaction cost [16,41,42]. However, today the landscape has changed.

The subprime mortgage crisis (2007–2009) recalled the limits of mainstream finance and reinforced the investigation of new or alternative models [40].

The probabilists has proved that as soon as there exist very small transaction costs, then arbitrage with fBm does no more exist [26,27]. Recent research in finance investigates pricing for fBm, see [19].

Then, modeling stock prices by mBm has been proposed. [24] pointed that there exists no arbitrage opportunity for such a mBm with a random time-varying Hurst index.

Behavioral finance provides economic explanations: Market efficiency corresponds to $H (t) = 1 / 2$ . For periods where $H (t) < 1 / 2$ the market overreacts, which means in probabilistic term, that the increments of the (log)price process are negatively correlated (anti-persistence), whereas for periods where $H (t) > 1 / 2$ the market under-reacts meaning that the increments of the price are positively correlated (persistence) [13–15,23]. For example, [25] notes that crashes on the Polish stock exchange market were following period where $H (t) < 1 / 2$ .

To sum up, new probabilist results show that fBm is an admissible model with transaction costs, statistics shows that mBm better fit the data and behavioral finance explains that under-reaction (

H (t) > 1 / 2

) is due to overconfidence of investors, whereas overreaction (

H (t) < 1 / 2

) is due to loss of confidence.

The next logical step for probability theorists would be to assume that the Hurst index is itself a stochastic process, that is to say with irregular paths, as the multifractional process with random exponent (MPRE) [5]. However, we show in this paper that this choice is counter productive as this complex model results from a statistical artifact.

We apply the test on financial data. The first application concerns the price of the NASDAQ (National Association of Securities Dealers Automated Quotations) of the US equity market (Fig. 16). We then aim at estimating the time varying Hurst index function for the NASDAQ with different estimators (Fig. 17): A piecewise constant function in magenta (see [8]), a mBm in green and a polynomial function of degree 9 in blue.

Fig. 16.

Daily NASDAQ spot price from February 1971 to September 2009.

Fig. 17.

Different estimators for Hurst index for daily NASDAQ spot price.

The second application concerns the daily prices of West Texas Intermediate oil (WTI) from January 1986 until May 2016 (Fig. 18). We estimate Hurst index with the same estimators used in the precedent example (Fig. 19). For the oil price, we reject the second and the third estimator and we accept the model with a constant estimator with $\tilde{H} (t) = 0.5052$ .

Fig. 18.

Daily price of the oil WTI.

Fig. 19.

Three estimators for Hurst index for daily WTI price.

Two considered series illustrate time-varying Hurst index in financial market. They show very different patterns of the prices’ persistence.

(1)

The observation of the H index on the Nasdaq market highlights two main facts:

A downward trend of the index, that goes from $H = 0.75$ , during the emergence of the market in the 1970s, to $H = 0.5$ . This fact can be interpreted as a form of a convergence towards the market efficiency that probably stems from its greater maturity. In other words, the market would penalize more the misvaluation of financial assets than in the years before 2001, when the technology bubble has burst.

A cyclical evolution of the H index, namely the recurrence of periods of market overreaction and under-reaction, which can probably be interpreted as the existence of misvaluation, at least in some segments of the market.

(2)

The most remarkable fact with regard to the crude oil market, is the cyclical nature of the H index around its efficiency value and the absence of any trend. Most of the studies do no reject the weak efficiency hypothesis. For instance, the results are in accordance with [3] which shows, using daily record of oil prices, that the crude oil market could be described by a random-walk process when short time scales are considered. Nevertheless, the topic is still a matter of debate, see e.g. [2].

6. Conclusion

The naive multifractional estimator ${\hat{H}}_{n} (t)$ is too complicated and has too many fluctuations that appear as a statistical artifact as shown in Figs 1 and 2. Moreover, we have built a fitting test which asymptotically rejects ${\hat{H}}_{n} (t)$ as an appropriate estimator of the time-varying right Hurst index $H (t)$ . Next, this fitting test is used to select the simplest time-varying Hurst index $\tilde{H} (t)$ from a given family of models, by a Portemanteau procedure. We have also proposed a family of piecewise polynomial functions. Indeed, the Hurst exponent could be random without being itself a stochastic process. For instance a piecewise affine with change of slope at random times is still a random exponent. So, we have disentangled a random time-varying Hurst exponent from a rough oscillating exponent resulting from a statistical artifact. Furthermore, the selected model is simpler and fits better the right value of the Hurst index $H (t)$ . Consequently, this study opens the way to further research like on line detection of change of slope of the Hurst index or study of the different kinds of families of model.

Footnotes

Acknowledgement

This research was supported by grant ANR-12-BS01-0016-01 “Do Well B.” (2012–2016) and by the project PEPS CNRS and Clermont-Ferrand Universities “Monitoring des états de stress”.

Proof of our main result – Theorem 3.2

For the sake of completeness, we recall the well-known lemma:

Preliminary remark. From Theorem 2.3 [17,18], we have the CLT (2.4). As pointed by Bardet–Surgailis [7], see e.g. Theorem 2.4, we also get a CLT for finite dimensional distribution, that is to say that for all finite family of times ${(t_{ℓ})}_{ℓ = 1, \dots, L}$ , we get $\begin{matrix} (A.1) & \begin{matrix} \sqrt{2 ε_{n} \cdot n} \times ({\hat{H}}_{n}^{(GQV)} (t_{ℓ}) - H (t_{ℓ})) \\ \overset{[}{} n \to \infty] D ⟶ G^{'} (t_{ℓ}), \end{matrix} \end{matrix}$ where $G^{'}$ is a zero mean Gaussian process with a covariance structure given by (2.5), (2.6).

Step 1. Let $G^{'}$ be a zero mean Gaussian process with a covariance structure given by (2.5), (2.6), and set $\begin{matrix} V_{L} : = \frac{1}{L} [\sum_{ℓ = 1}^{L} {| G^{'} (t_{ℓ}) |}^{2}] . \end{matrix}$ We will prove in this step that a CLT is in force for $V_{L}$ when $L \to \infty$ . Since $G^{'}$ is a zero mean Gaussian process, we can mimic the strategy used in [4]. In this frame, a sufficient condition to have a CLT for $V_{L}$ is $\begin{matrix} (A.2) & \begin{matrix} \frac{{max}_{ℓ = 1, \dots, L} \sum_{k = 1, \dots, L} | cov (G^{'} (t_{k}), G^{'} (t_{ℓ})) |}{\sqrt{var (V_{L})}} \\ \to 0 as L \to \infty . \end{matrix} \end{matrix}$ By using Lemma A.1, we get the following calculation $\begin{array}{rcl} var (V_{L}) & = & var (\sum_{k = 1}^{L} {| G^{'} (t_{k}) |}^{2}) \\ = & \sum_{k_{1}, k_{2} = 1}^{L} cov (G^{'} {(t_{k_{1}})}^{2}, G^{'} {(t_{k_{2}})}^{2}) \\ = & 2 \sum_{k_{1}, k_{2} = 1}^{L} cov {(G^{'} (t_{k_{1}}), G^{'} (t_{k_{2}}))}^{2} \\ ⩾ & 2 \sum_{k_{1} = k_{2} = 1}^{L} cov {(G^{'} (t_{k_{1}}), G^{'} (t_{k_{2}}))}^{2} \\ = & 2 \sum_{k = 1}^{L} var {(G^{'} (t_{k}))}^{2} \\ ⩾ & 2 L min_{k = 1, \dots, L} γ {(H (t_{k}))}^{2} \\ ⩾ & (2 L) \times min_{H \in (0, 1)} γ {(H)}^{2} . \end{array}$ On the other hand, formulas (2.5), (2.6) imply $\begin{array}{rcl} \sum_{k = 1, \dots, L} | cov (G^{'} (t_{k}), G^{'} (t_{ℓ})) | & = & | var (G^{'} (t_{ℓ})) | \\ = & | γ (H (t_{ℓ})) | \\ ⩽ & max_{H \in (0, 1)} | γ (H) | . \end{array}$ Therefore, we get $\begin{array}{l} \frac{{max}_{ℓ = 1, \dots, L} \sum_{k = 1, \dots, L} | cov (G^{'} (t_{k}), G^{'} (t_{ℓ})) |}{\sqrt{var (V_{L})}} \\ ⩽ \frac{1}{\sqrt{2 L}} \times \frac{{max}_{H \in (0, 1)} | γ (H) |}{{min}_{H \in (0, 1)} | γ (H) |}, \end{array}$ which implies (A.2). To sum up, we have $\begin{array}{rcl} V_{L} & = & μ_{L} + S_{L} \times ξ_{L}, \end{array}$ where $μ_{L} = E (V_{L})$ is the expected value, $S_{n} = \sqrt{Var (V_{L})}$ is the standard deviation, and $\begin{array}{rcl} ξ_{L} & \overset{[}{} L \to \infty] D ⟶ & N (0; 1) . \end{array}$ Since $E (V_{L}) = \frac{1}{L} \sum_{ℓ = 1}^{L} γ (H (t_{ℓ}))$ and $Var (V_{L}) = \frac{2}{L} \sum_{ℓ = 1}^{L} γ {(H (t_{ℓ}))}^{2}$ , we get $\begin{array}{rcl} ξ_{L} & = & \frac{\frac{1}{L} \sum_{ℓ = 1}^{L} [| G^{'} (t_{ℓ}) |^{2} - γ (H (t_{ℓ}))]}{\sqrt{\frac{2}{L} \sum_{ℓ = 1}^{L} γ {(H (t_{ℓ}))}^{2}}} \\ \overset{[}{} L \to \infty] D ⟶ & N (0; 1) . \end{array}$ By Dini Theorem, the convergence in distribution implies the uniform convergence of the cumulative distribution functions. Therefore, $\begin{array}{rcl} (A.3) & \begin{matrix} \forall ε > 0, \exists L_{0} such that \\ \forall L > L_{0}, | F_{ξ_{L}} (x) - F (x) | ⩽ ε / 2, \\ \forall x \in R, \end{matrix} \end{array}$ where F denotes the cumulative distribution function of the standard zero mean Gaussian variable, and $F_{ξ_{L}}$ the one of the variable $ξ_{L}$ .

Step 2. For every $L > L_{0}$ , Theorem 2.4 induces that $\begin{array}{rcl} Γ_{L, n} & : = & \frac{1}{L} \sum_{ℓ = 1}^{L} [2 (ε_{n} \cdot n) \\ \times {| {\hat{H}}_{n}^{(GQV)} (t_{ℓ}) - H (t_{ℓ}) |}^{2} \\ - γ (H (t_{ℓ}))] \\ / \sqrt{\frac{2}{L} \sum_{ℓ = 1}^{L} γ {(H (t_{ℓ}))}^{2}} \\ \overset{[}{} n \to \infty] D ⟶ & \frac{\frac{1}{L} \sum_{ℓ = 1}^{L} [| G^{'} (t_{ℓ}) |^{2} - γ (H (t_{ℓ}))]}{\sqrt{\frac{2}{L} \sum_{ℓ = 1}^{L} γ {(H (t_{ℓ}))}^{2}}} = ξ_{L} . \end{array}$ This induces that for all $L > L_{0}$ , there exists $N_{0} (L)$ such that for all $n > N_{0} (L)$ , $\begin{array}{rcl} (A.4) & | F_{Γ_{L, n}} (x) - F_{ξ_{L}} (x) | & ⩽ & ε / 2, \forall x \in R . \end{array}$ By combining (A.3) and (A.4), we can deduce that $\begin{array}{rcl} | F_{Γ_{L, n}} (x) - F (x) | & ⩽ & ε, \forall x \in R . \end{array}$ This finishes the proof of Theorem 3.2. □

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